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WORKS    OF    PROF.  WM.  H.    BURR 

PUBLISHED    BY 

JOHN  WILEY  &  SONS,  Inc. 


Ancient  and  Modern  Engineering  and  the  Isthmian 
Canal. 

xv +476  pages,  6  by  9,  profusely  illustrated,  including 
many  half-tones.  Cloth,  $3.50  net.  • 

Elasticity    and    Resistance   of    Materials  of    Engi- 
neering. 

For  the  use  of  Engineers  and  Students.  Containing  the 
latest  engineering  experience  and  tests.  Seventh  Edi- 
tion, revised.  946  pages,  6  by  9.  Cloth,  $5.50  net. 

Suspension  Bridges — Arch  Ribs  and  Cantilevers. 

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folding  plates.  Cloth,  $4.50  net. 


BY  PROF.  BURR  and  DR.  FALK 

The  Graphic  Method  by  Influence  Lines  for  Bridge 
and  Roof  Computations. 

xi  4-253  pages,  6  by  9,  4  folding  plates.      Cloth,  $3.00. 
The  Design  and  Construction  of  Metallic  Bridges. 

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BY  DR.  MYRON  S.  FALK 

PUBLISHED    BY 

MYRON   C.    CLARK   PUBLISHING   CO., 
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Cements,  Mortars  and  Concretes— Their  Physical 
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Containing  the  results  of  late  investigations  upon  these 
materials.      176  pages.  6  by  9.      Cloth,  $2.50. 


' 


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THE 


ELASTICITY  AND  RESISTANCE 


OF  THE 


MATERIALS  OF  ENGINEERING. 


BY 

WM.    H.    BURR,    C.E., 

PROFESSOR  OF   CIVIL   ENGINEERING  IN   COLUMBIA   UNIVERSITY   IN  THE   CITY   OF   NEW  YORK? 

CONSULTING  ENGINEER;  MEMBER  OF  THE  AMERICAN  SOCIETY  OF  CIVIL  ENGINEERS; 

MEMBER  OF  THE  INSTITUTION  OF  CIVIL  ENGINEERS   OF 

GREAT   BRITAIN. 


SEVENTH  EDITION,  THOROUGHLY  REVISED 
TOTAL   ISSUE,    SEVEN   THOUSAND 


NEW  YORK 

JOHN  WILEY  &  SONS,  INC. 
LONDON:   CHAPMAN  &   HALL,  LIMITED 


Copyright,  1883,  1903 
WM.  H.  BURR. 


Copyright  renewed,  191  ir 

BY 
WM.  H.  BURR. 

•333831 


,  i9is,        -r*  A 

TA 

S3 


PREFACE  TO   SEVENTH   EDITION. 


THE  rapid  development  which  has  characterized  all 
branches  of  engineering  construction  during  the  past 
decade  carries  with  it  corresponding  advances  in  experi- 
mental and  analytic  work  in  that  field  of  engineering 
science  known  as  the  Elasticity  and  Resistance  of  Mate- 
rials. In  the  present  edition  of  this  s'book,  prepared  to 
meet '  the  advancing  requirements  of  the  profession,  it 
will  be  observed  that  much  of  the  older  matter  has  been 
canceled  and  displaced  by  many  new  topics  now  become 
of  practical  importance,  so  that  new  material  constitutes 
probably  not  less  than  three-quarters  of  the  volume.  These 
new  parts  will  readily  be  discovered  by  a  glance  at  the 
contents.  It  may  be  well,  however,  to  state  that  the 
treatment  of  reinforced  concrete,  the  general  analysis  of 
which  as  a  development  of  the  common  theory  of  flexure 
was  first  given  in  a  prior  edition  of  this  book,  has  been 
extended  to  cover  substantially  all  the  principal  features 
of  that  special  field.  The  analysis  given  is  general, 
but  simple  and  free  from  the  superfluous  and  labor- 
increasing  accretions  which,  for  some  not  obvious  reasons, 
have  found  place  in  some  of  the  commonly  used  formulae. 
Results  of  the  most  recent  experimental  investigations 
have  been  used  for  the  requisite  empirical  data,  so  as  to 
make  the  book  a  real  work  on  the  Elasticity  and  Resist- 
ance of  the  Materials  of  Engineering  rather  than  a  mere 
matter  of  applied  mechanics. 

W.  H.  B. 
COLUMBIA  UNIVERSITY, 
Oct.  i,  1915. 


333831 


CONTENTS. 


PART   I. 

ANALYTICAL. 

CHAPTER  I. 

ELEMENTARY  THEORY  OF  ELASTICITY  IN  AMORPHOUS  SOLID 

BODIES. 

ART.  PAGE 

1 .  General  Statements I 

2.  Coefficient  or  Modulus  of  Elasticity 4 

3.  Direct  Stresses  of  Tension  and  Compression 7 

4.  Lateral  Strains 9 

5.  Relation  between  the  Coefficients  of  Elasticity  for  Shearing  and 

Direct  Stress  in  a  Homogeneous  Body 1 1 

6.  Shearing  Stresses  and  Strains 13 

7.  Relation   between   Moduli   of   Elasticity  and   Rate   of   Change   of 

Volume 18 

8.  All  Stresses  Parallel  to  One  Plane — Resultant  Stress  on  any  Plane 

Normal  to  the  Plane  of  Action  of  the  Stresses 21 

Sum  of  Normal  Components 24 

9.  The    Ellipse    of    Stress — Greatest    Intensity    of    Shearing    Stress — 

Equivalence  of  Pure  Shear  to  Two  Principal  Stresses  of  Opposite 
Kinds   but   Equal   Intensities — Greatest   Obliquity  of  Resultant 

Stress  on  any  Plane 26 

Greatest  Intensity  of  Shearing  Stress 29 

Equivalence  of  Pure  Shear  to  Two  Principal  Stresses  of  Opposite 

Kinds  but  Equal  Intensities 30 

Greatest  Obliquity  of  Resultant  Stress  on  any  Plane 31 

10.  Ellipse  of  Stress  and  Resulting  Formula  for  the  Special  Case  of 

Zero  Intensity  of  One  of  the  Known  Direct  Stresses 33 

11.  General  Condition  of  Stress— Ellipsoid  of  Stress 36 

Principal  Stresses  and  Ellipsoid  of  Stress 40 

vii 


viii  CONTENTS. 


ART. 


PAGE 


12.  Ellipse  and  Ellipsoid  of  Strain 43 

13.  Orthogonal  Stresses .-., 43 

CHAPTER  II. 

FLEXURE. 

'14.  The  Common  Theory  of  Flexure 49 

15.  The  Distribution  of  Shearing  Stress  in  the  Normal  Section  of  a  Bent 

Beam 57 

Distribution  of  Shear  in  Circular  and  Other  Sections 62 

1 6.  External  Bending  Moments  and  Shears  in  General 64 

17.  Intermediate  and  End  Shears 68 

1 8.  Maximum  Reactions  for  Bridge  Floor  Beams •. 74 

19.  Greatest  Bending  Moment  Produced  by  Two  Equal  Weights 76 

20.  Position  of  Uniform  Load  for  Greatest  Shear  and  Greatest  Bending 

Moment   at  any  Section    of   a  Non-continuous   Beam — Bending 

Moments  of  Concentrated  Loads 79 

21.  Greatest  Bending  Moment  in  a  Non-continuous  Beam  Produced  by 

Concentrated  Loads 83 

22.  Moments  and  Shears  in  Special  Cases 94 

Case  1 95 

Case  II 96 

Case  III 98 

23.  Recapitulation  of  the  General  Formulae  of  the   Common    Theory   of 

Flexure 99 

24.  The  Theorem  of  Three  Moments 102 

25.  Short  Demonstration  of  the   Common   Form   of   the   Theorem  of 

Three  Moments 114 

26.  Reaction  under  Continuous  Beam  of  any  Number  of  Spans 1 1 8 

27.  Deflection  by  the  Common  Theory  of  Flexure 121 

Deflection  Due  to  Shearing 125 

28.  The  Neutral  Curve  for  Special  Cases 126 

Case  I 126 

Case  II 129 

Case  III 131 

Addendum  to  Art.  28 143 

29.  Direct  Demonstration  for  Beam  Fixed  at  One  End  and  Simply  Sup- 

ported at  the  Other  under  Uniform  and  Single  Loads 144 

Special  Case,  a  =  \ 149 

30.  Direct  Demonstration  for  Beams  Fixed  at  Both  Ends  under  Uniform 

and  Single  Loads : 150 

31.  Deflection  Due  to  Shearing  in  Special  Cases 153 

32.  The  Common  Theory  of  Flexure  for  a  Beam  Composed  of  Two 

Materials 5 


CONTENTS.  ix 

ART.  PAGE 

33.  Graphical  Determination  of  the  Resistance  of  a  Beam 160 

34.  Greatest  Stresses  at  any  Point  in  a  Beam 162 

35.  The  Flexure  of  Long  Columns 169 

36.  Special  Cases  of  Flexure  of  Long  Columns .  .  . .  . 175 

Flexure  by  Oblique  Forces 175 

Column  Free  at  Upper  End  and  Fixed  Vertically  at  Lower  End 
with  either  Inclined  or  Vertical  Loading  at  Upper  End 177 

CHAPTER  III. 
TORSION. 

37.  Torsion  in  Equilibrium 182 

Twisting  Moment  in  Terms  of  Horse-power  II 188 

Hollow  Circular  Cylinders 189 

38.  Practical  Applications  of  Formulae  for  Torsion 190 

Steel 190 

Wrought  Iron 192 

Cast  Iron 192 

Alloys  of  Copper,  Tin,  Zinc  and  Aluminum 193 

Other  Sections  than  Circular 196 

CHAPTER  IV. 
HOLLOW  CYLINDERS  AND  SPHERES. 

39.  Thin  Hollow  Cylinders  and  Spheres  in  Tension 197 

40.  Thick  Hollow  Cylinders 203 

Case  of  Exterior  Pressure  Greater  than  Interior  Pressure 211 

41.  Radial    Strain    or    Displacement    in    Thick     Hollow     Cylinders — 

Stresses  Due  to  Shrinkage  of  One  Hollow  Cylinder  on  Another.  .  .  212 

Radial  Strain  or  Displacement 212 

Stresses  Due  to  Shrinkage 213 

Inner  Cylinder  in  Compression 217 

Outer  Cylinder  in  Tension 218 

,     Combined  Cylinder  under  High  Internal  Pressure 219 

42.  Thick  Hollow  Spheres , 224 

Radial  Displacement  at  any  Point  in  the  Spherical  Shell 230 

CHAPTER  V. 

RESILIENCE. 

43.  General  Considerations 231 

44.  The  Elastic  Resilience  of  Tension  and  Compression  and  of  Flexure.    232 

The  Resilience  of  Bending  or  Flexure 233 


X  CONTENTS. 

ART.  PAGE 

The  Resilience  Due  to  the  Vertical  or  Transverse  Shearing  Stresses 

in  a  Bent  Beam " 236 

The  Total  Resilience  Due  to  Both  Direct  and  Shearing  Stresses .  .  .  239 

45.  Resilience  of  Torsion 240 

46.  Suddenly  Applied  Loads 242 


CHAPTER  VI. 
COMBINED  STRESS  CONDITIONS. 

47.  Combined  Bending  and  Torsion 246 

First  Method 248 

Second  Method 250 

48.  Combined  Bending  and  Direct  Stress 254 

49.  The  Eye-bar  Subjected  to  Bending  by  Its  Own  Weight  or  Other 

Vertical  Loading 255 

Approximate  Method 256 

50.  The  Approximate  Method  Ordinarily  Employed 258 

5 1 .  Exact  Method  of  Treating  Combined  Bending  and  Direct  Stress. .  . .  263 

52.  Combined  Bending  and  Direct  Stress  in  Compression  Members 268 

Exact  Method  for  Combined  Compression  and  Bending 271 


PART  II. 
TECHNICAL. 

CHAPTER  VII. 
TENSION. 

53.  General  Observations. — Limit  of  Elasticity. — Yield  Point 281 

Yield  Point 284 

54.  Ultimate  Resistance 285 

55.  Ductility — Permanent  Set 286 

56.  Cast  Iron 286 

Modulus  of  Elasticity  and  Elastic  Limit 286 

Resilience,  or  Work  Performed  in  Straining  Cast  Iron 290 

Ultimate  Resistance 292 

Effects  of  Remelting,  Continued  Fusion,  Repetition  of  Stress,  and 

High  Temperature 294 


CONTENTS.  xi 

ART.  PAGE 

57.  Wrought   Iron — Modulus   of   Elasticity — Limit   of   Elasticity    and 

Yield  Point— Resilience — Ultimate  Resistance  and  Ductility 295 

Modulus  of  Elasticity 296 

Limit  of  Elasticity  and  Yield  Point  Resilience 297 

Ductility  and  Resilience 299 

Ultimate  Resistance 301 

Ductility. 302 

Fracture  of  Wrought  Iron 302 

58.  Steel 303 

Modulus  of  Elasticity 303 

Variation  of  Ultimate  Resistance  with  Area  of  Cross-section 308 

Influence  of  Shortness  of  Specimen 309 

Elastic  Limit,  Resilience,  and  Ultimate  Resistance 310 

Shape  Steel  and  Plates 315 

Carbon  Steel  for  Towers 318 

Carbon  Steel  for  Suspended  Structures 319 

Nickel  Steel  for  Stiffening  Trusses 319 

Steel  Wire 320 

Steel  Castings 321 

Rail  Steel 323 

Rivet  Steel 324 

Nickel  Steel 325 

Vanadium  Steel 328 

Effect  of  Low  and  High  Temperatures 333 

Hardening  and  Tempering 336 

Annealing 338 

Effect  of  Manipulations  Common  to  Constructive  Processes; 

also  Punched,  Drilled  and  Reamed  Holes 339 

Change  of  Ultimate  Resistance,  Elastic  Limit  and  Modulus  of 

Elasticity  by  Retesting 342 

Fracture  of  Steel 343 

The  Effects  of  Chemical  Elements  on  the  Physical  Qualities  of 

Steel 343 

59.  Copper,  Tin,  Aluminum,    and  Zinc,    and   Their  Alloys — Alloys   of 

Aluminum — Phosphor-Bronze — Magnesium 346 

Ultimate  Resistance  and  Elastic  Limit 348 

Alloys  of  Aluminum 352 

Alloys  of  Aluminum  and  Copper 357 

Bronzes  and  Brass  Used  by  the  Board  of  Water  Supply  of  New 

York  City 359 

Phosphor-Bronze . 361 

Bauschinger's  Tests  of  Copper  and  Brass  as  to  Effect  of  Repeated 
Application  of  Stress 361 


xii  CONTENTS. 

ART.  PAGE 

60.  Cement,  Cement  Mortars,  etc. — Brick 362 

Modulus  of  Elasticity .  .< 363 

Ultimate  Resistance 365 

Weight  of  Concrete 372 

Adhesion  between  Bricks  and  Cement  Mortar 373 

The  Effect  of  Freezing  Cements  and  Cement  Mortars 375 

The  Linear  Thermal  Expansion  and  Contraction  of  Concrete  and 

Stone 377 

61.  Timber  in  Tension 379 


CHAPTER  VIII. 
COMPRESSION. 

62.  Preliminary 385 

63.  Wrought  Iron 387 

Modulus  of  Elasticity 387 

Limit  of  Elasticity  and  Ultimate  Resistance 388 

64.  Cast  Iron 388 

65.  Steel 389 

66.  Copper,  Tin,  Zinc,  Lead,  and  Alloys 391 

67.  Cement — Cement  Mortar — Concrete 395 

68.  Bricks  and  Brick  Piers 409 

Brick  Piers 413 

69.  Natural  Building  Stones 420 

70.  Timber 426 


CHAPTER  IX. 

RIVETED  JOINTS  AND  PIN  CONNECTION. 

7 1 .  Riveted  Joints 435 

Kinds  of  Joints 435 

72.  Distribution  of  Stress  in  Riveted  Joints 437 

Bending  of  the  Plates 437 

Net  Section  of  Plates 439 

Bending  of  the  Rivets 440 

The  Bearing  Capacity  of  Rivets 441 

Bending  of  Plate  Metal  in  Front  of  Rivets 442 

Shearing  of  Rivets 443 


CONTENTS.  xiii 


ART. 

73.  Diameter  and  Pitch  of  Rivets  and  Overlap    of    Plate.—  Distance 

between  Rows  of  Riveting  ............  .......................   445 

Diameter  of  Rivets  .........................................  445 

Pitch  of  Rivets  ............................................   446 

Overlap  of  Plate  ...........................................  447 

Distance  between  Rows  of  Riveting  ...........................   448 

74.  Lap-joints,    and    Butt-joints    with     Single    Butt-strap    for    Steel 

Plates  ........................................................   448 

75.  Steel  Butt-joints  with  Double  Cover-plates  ......................   452 

76.  Tests  of  Full-size  Riveted  Joints  .................  '.  ..............   454 

Efficiencies  ...............................................   461 

77.  Tests  of  Joints  for  the  American   Railway  Engineering   and  Main- 

tenance of  Way  Association  and    for  the  Board  of  Consulting 
Engineers  of  the  Queb-    Bridge  ..............................   462 

Friction  of  Riveted  Joints  ........  .  ..........................   465 

78.  Riveted  Truss  Joints  ..........................................   467 

Diagonal  Joints  ...........................................  469 

Riveted  Joints  in  Angles  ...................  .  ................  469 

Hand  and  Machine  Riveting  ................................  470 

79.  Welded  Joints  .......................................  .........  47° 

80.  Pin  Connections  ,.,,,,,,,,,,  ..................................  47° 


CHAPTER  X. 
LONG  COLUMNS. 

81.  Preliminary  Matter 474 

Principal  Moments  of  Inertia 477 

82.  Gordon's  Formula  for  Long  Columns 481 

83.  Tests  of  Wrought-iron  Phoenix  Columns,  Steel   Angles   and   Other 

Steel  Columns 490 

Steel  Columns 496 

Typical  Formula  Now  in  Use 503 

Details  of  Columns 505 

84.  Complete  Design  of  Pin-end  Steel  Columns 506 

85.  Cast-iron  Columns 520 

86.  Timber  Columns 528 

Formula  of  C.  Shaler  Smith 531 

Tests  of  White  Pine  and  Yellow  Pine  Full-size  Sticks  with  Flat 
Ends 533 


xiv  CONTENTS. 

CHAPTER  XI. 
SHEARING  AND  TORSION. 

ART.  PAGE 

87.  Modulus  of  Elasticity 540 

88.  Ultimate  Resistance '  543 

Wrought  Iron 543 

Cast  Iron 544 

Steel 545 

Copper,  Tin,  Zinc,  and  Their  Alloys 546 

Timber 547 

Natural  Stones 549 

Bricks 550 

CHAPTER  XII. 
BENDING  OR  FLEXURE. 

89.  Modulus  of  Elasticity 552 

90.  Formulae  for  Rupture 552 

91.  Beams  with  Rectangular  and  Circular  Sections 554 

High  Extreme  Fibre  Stress  in  Short  Solid  Beams 556 

Steel 558 

Cast  Iron 560 

Alloys  of  Aluminum -. 560 

Copper,  Tin,  Zinc,  and  their  Alloys 561 

Timber  Beams 563 

Failure  of  Timber  Beams  by  Shearing  along  the  Neutral  Surface.  571 

Influence  of  Time  on  the  Strains  of  Timber  Beams 574 

Concrete  Beams 575 

Natural-stone  Beams 586 

CHAPTER  XIII. 
CONCRETE-STEEL  MEMBERS. 

92.  Composite  Beams  or  Other  Members  of  Concrete  and  Steel 588 

93.  Physical  Features  of  the  Concrete-steel  Combination  in  Beams 589 

94.  Rate  at  which  Steel  Reinforcement  Acquires  Stress 592 

95.  Ultimate  and  Working  Values  of  Empirical  Quantities  for  !Concrete- 

steel  Beams 598 

96.  General  Formulae  and  Notation  for   the   Theory  of   Concrete-steel 

Beams  according  to  the  Common  Theory  of  Flexure 600 


CONTENTS.  xv 

ART.  PAGE 

97.  T-beams  of  Reinforced  Concrete 604 

Position  of  Neutral  Axis 605 

Balanced  or  Economic  Steel  Reinforcement 608 

Formulae,  to  Locate  Neutral  Axis  in  T-beams 610 

98.  Bending  Moments  in  Concrete-steel  T-beams  by  Common    Theory 

of  Flexure 614 

Neglect  of  Concrete  in  Tension 615 

Special  Case  of  Neutral  Axis  in  under  Surface  of  Flange 616 

99.  Concrete  Steel  Beams  of  Rectangular  Section 616 

Formula  to  Locate  Neutral  Axis  in  Beams  of  Rectangular  Section  616 

Bending  Moments  for  Rectangular  Sections 618 

Neglect  of  Concrete  in  Tension 619 

100.  Shearing  Stresses  and  Web  Reinforcements  in  Reinforced  Concrete 

Beams 620 

101.  Working  Stresses  and  Other  Conditions  in  Reinforced  Concrete 

Design — Design  of  T-beams 629 

Working  Stresses 631 

Working  Compression  in  Extreme  Fibre  of  Beam 631 

Shear  and  Diagonal  Tension 632 

Bond  or  Adhesive  Shear 633 

Steel  Reinforcement 633 

Modulus  of  Elasticity 633 

Design  of  T-beam  for  Heavy  Uniform  Load 634 

Design  of  Continuous  Floor  Slab  for  6-foot  Spans 639 

102.  Reinforced  Concrete  Columns 641 

Lateral  Reinforcement  and  Shrinkage 642 

Longitudinal  Reinforcement 644 

Types  of  Columns 646 

Working  Stresses 650 

103.  Division  of  Loading  between  the  Concrete  and    Steel  under  the 

Common  Theory  of  Flexure 655 


CHAPTER  XIV. 
ROLLED  AND  CAST-FLANGED  BEAMS. 

104.  Flanged  Beams  in  General 659 

105.  Flanged  Beams  with  Unequal  Flanges 66 1 

106.  Flanged  Beams  with  Equal  Flanges 665 

107.  Rolled  Steel  Flanged  Beams 669 

108.  The  Deflection  of  Rolled  Steel  Beams 677 

109.  Wrought-iron  Rolled  Beams 679 


xvi  CONTENTS. 

CHAPTER  XV. 
PLATE  GIRDERS. 

ART.  PAGE 

1 10.  The  Design  of  a  Plate  Girder 683 

Bending  Moments 685 

Shears 689 

Web  Plate 692 

Flanges 693 

Length  of  Cover-plates 695 

Pitch  of  Rivets  in  Flanges '.  .  .  .   698 

Pitch  of  Rivets  in  Cover-plates 702 

Top-flange 7°3 

End  Stiffeners 7°3 

Intermediate  Stiffeners 705 

Splices  in  Flanges 705 

Splices  in  Web  Plates 706 

General  Considerations 7°7 

in.  Length  of  Cover- plates 7°8 

1 12.  Pitch  of  Rivets 710 


CHAPTER  XVI. 
MISCELLANEOUS  SUBJECTS. 

113.  Curved  Beams  in  Flexure 712 

1 14.  Stresses  in  Hooks 719 

115.  Eccentric  Loading 725 

Rotation  of  the  Neutral  Axis  about  a  Fixed  Point  in  It 730 

Application  of  Preceding  Procedures  to  Z-bar  and  Rectangular 

Sections 730 

General  Observations 734 

1 1 6.  General  Flexure  Treated  by  the  Core  Method 735 

Component  Moments 738 

117.  Planes  of  Resistance  in  Oblique  or  General  Flexure 739 

1 18.  Deflection  in  Oblique  Flexure 745 

119.  Elastic   Action   under   Direct   Loading  of  a   Composite   Piece   of 

Material 749 

120.  Helical  Spiral  Springs 750 

Small  Pitch  Angle 758 

Rotation  of  Spring  Prevented. 759 

Axial  Extension  or  Compression  Prevented 759 

Work  Performed  in  Distorting  the  Spring 760 


CONTENTS.  xvii 

ART.  PAGE 

121.  Plane  Spiral  Springs 761 

122.  Problems 762 

123.  Flat  Plates ; 765 

Square  Plates —  Uniform  Load 766 

Square  Plates — Single  Center  Load 767 

Rectangular  Plates —  Uniform  Load 768 

Circular  Plate — Uniform  Load — Center  Load. 769 

Elliptical  Plates — Centre  Load — Uniform  Load 771 

Flat  Plates  Fixed  at  Edges 772 

124.  Resistance  of  Flues  to  Collapse 774 

125.  Approximate  Treatment  of  Solid  Metallic  Rollers 778 

126.  Resistance  to  Driving  and  Drawing  Spikes 781 

127.  Shearing  Resistance  of  Timber  behind  Bolt  or  Mortise  Holes 786 

128.  Method  of  Least  Work — Stresses  in  a  Bridge  Portal 788 

Stresses  in  a  Bridge  Portal 789 


CHAPTER  XVII. 
THE  FATIGUE  OF  METALS. 

129.  Woehler's  Law 795 

130.  Experimental  Results 796 

131.  Formulae  of  Launhardt  and  Weyrauch 801 

132.  Influence  of  Time  on  Strains 805 


CHAPTER  XVIII. 
THE  FLOW  OF  SOLIDS. 

133.  General  Statements 809 

134.  Tresca's  Hypotheses 811 

135.  The   Variable    Meridian    Section   of    the    Primitive   Central    Cyl- 

inder    813 

136.  Positions  in  the  Jet  of  Horizontal  Sections  of  the  Primitive  Central 

Cylinder 815 

137.  Final  Radius  of  a  Horizontal    Section  of  the  Primitive  Central 

Cylinder 817 

138.  Path  of  Any  Molecule 817 


xviii  CONTENTS. 


APPENDIX  I. 

ELEMENTS  OF   THEORY   OF   ELASTICITY   IN  AMORPHOUS 
SOLID   BODIES. 


CHAPTER  I. 
GENERAL  EQUATIONS. 

ART.  PAGE 

1.  Expressions  for  Tangential  and  Direct  Stresses  in  Terms  of  the  Rates 

of  Strains  at  Any  Point  of  a  Homogeneous  Body 820 

2.  General  Equations  of  Internal  Motion  and  Equilibrium 826 

3.  Equations  of  Motion  and  Equilibrium  in  Semi-polar  Co-ordinates. .  .  832 

4.  Equations  of  Motion  and  Equilibrium  in  Polar  Co-ordinates 839 


CHAPTER  II. 
THICK,  HOLLOW  CYLINDERS  AND  SPHERES,  AND  TORSION. 

5.  Thick,  Hollow  Cylinders 847 

6.  Torsion  in  Equilibrium 853 

Equations  of  Condition  in  Rectangular  Co-ordinates 860 

Solutions  of  Eqs.  (13)  and  (21) 862 

Elliptical  Section  about  its  Centre 863 

Equilateral  Triangle  about  its  Centre  of  Gravity 866 

Rectangular  Section  about  an  Axis  Passing  through  its   Centre 

of  Gravity 869,  883 

Square  Section 878,  882 

Greatest  Intensity  of  Shear 880 

Circular  Section  about  its  Centre 884 

General  Observations 885 

7.  Torsional  Oscillations  of  Circular  Cylinders 886 

8.  Thick,  Hollow  Spheres 892 

CHAPTER  III. 

THEORY  OF  FLEXURE. 

9.  General  Formulae 897 


CONTENTS. 

APPENDIX   II. 

PAGE 
CLAVARINO'S   FORMULA 913 

APPENDIX   III. 

RESISTING   CAPACITY   OF  NATURAL    AND 

ARTIFICIAL    ICE 916 


INDEX 921 


. 


2  ELASTICITY  IN  AMORPHOUS  SOLID  BODIES.         [Ch.  I. 

These  stresses  and  strains  vary  in  character  according  to 
the  method  of  application  of  the  external  forces.  Each 
stress,  however,  is  accompanied  by  its  own  characteristic 
strain  and  no  other.  Thus  there  are  shearing  stresses  and 
shearing  strains,  tensile  stresses  and  tensile  strains,  com- 
pressive stresses  and  compressive  strains.  Usually  a 
number  of  different  stresses  with  their  corresponding 
strains  are  coexistent  at  any  point  in  a  body  subjected  to 
the  action  of  external  forces. 

It  is  a  matter  of  experience  that  strains  always  vary 
continuously  and  in  the  same  direction  with  the  corre- 
sponding stresses.  Consequently  the  stresses  are  con- 
tinuously increasing  functions  of  the  strains,  and  any 
stress  may  be  represented  by  a  series  composed  of  the 
ascending  powers  "(commencing  with  the  first)  of  the  strains 
multiplied  by  proper  coefficients.  When,  as  is  usually 
the  case,  the  displacements  are  very  small,  the  terms  of 
the  series  whose  indices  are  greater  than  unity  are  ex- 
ceedingly small  compared  with  the  first  term,  whose  index 
is  unity.  Those  terms  may  consequently  be  omitted 
without  essentially  changing  the  value  of  the  expression. 
Hence  follows  what  is  ordinarily  termed  Hooke's  law: 

The  ratio  between  stresses  and  corresponding  strains,  fot 
a  given  material,  is  constant. 

This  law  is  susceptible  of  very  simple  algebraic  repre- 
sentation. If  a  piece  of  material,  whose  normal  cross' 
section  is  -A,  is  subjected  to  either  tensile  or  compressive 
stress,  its  length  L  will  be  changed  by  the  amount  AL. 
If  P  be  the  external  force  or  loading  which  produces  that 
deformation  or  change  of  length,  the  amount  of  force  or 
stress,  supposed  to  be  uniformly  distributed,  acting  on  i 
square  inch  of  normal  cross-section  of  the  piece,  will  be 
found  by  dividing  the  total  force  P  by  the  area  of  cross- 
section  A.  This  amount  of  uniformly  distributed  stress 


Art.  i.]  GENERAL  STATEMENTS.  3 

is  called  the  '  '  intensity  of  stress,  '  '  and  it  is  a  most  impor- 
tant quantity.  In  dealing  with  the  effects  of  forces  or 
stresses  in  all  engineering  work,  the  amount  of  such  force 
Or  stress  on  a  square  unit  of  area,  usually  a  square  inch  in 
American  practice,  and  called  the  intensity,  is  often  the 
main  object  sought,  for  it  determines  the  question  whether 
material  is  carrying  too  much  or  too  little  load,  as  well  as 
many  other  related  questions. 

Again,  the  important  consideration  as  to  strain  is  the 
fractional  change  in  length  of  the  entire  piece,  and  not  the 
total  change  in  length  expressed  in  the  unit  adopted,  ordi- 
narily an  inch.  This  fractional  change  of  length  is  the  same 
as  the  amount  of  actual  change  of  each  linear  unit  of  the 
piece,  as  found  by  dividing  JL  by  L.  Inasmuch  as  that 
fraction  expresses  the  amount  of  change  in  length  for  each 
unit,  it  is  frequently  called  the  rate  of  change  of  length  or 
rate  of  deformation.  Hooke's  law  is  to  the  effect  that 
the  intensity  of  stress  is  proportional  to  the  rate  of  strain, 
and  its  analytic  expression  may  readily  be  written. 

Let  p  represent  the  intensity  of  any  stress  and  /  tbe 
strain  per  unit  of  length,  or,  in  other  words,  the  rate  of 
strain.  If  E  is  a  constant  coefficient,  Hooke's  law  will  be 
given  by  the  following  equation: 


If  the  intensity  of  stress  varies  from  point  to  point  of  a 
body,    Hooke's   law  may  be   expressed  by  the    following 

/-1-J-f-fo-rorrfi  a  1     t>n  notion  ' 


differential  equation : 


%-E.  W 


If  p  and  I   are  rectangular  coordinates,  eqs.   (i)  and  (2) 
are  evidently  equations  of  a  straight  line  passing  through 


4  ELASTICITY  IN  AMORPHOUS  SOLID  BODIES.         [Ch.  I. 

the  origin  of  coordinates.  It  will  hereafter  be  seen  that 
the  line  under  consideration  is  essentially  straight  for 
comparatively  small  strains  in  any  case,  and  for  some 
materials  it  has  no  straight  portions. 

Art.  2. — Coefficient  or  Modulus  of  Elasticity. 

In  general  the  coefficient  E  in  eq.  (i)  of  the  preced- 
ing article  is  called  the  coefficient  of  elasticity,  or,  more 
usually,  modulus  of  elasticity.  The  coefficient  of  elasticity 
varies  both  with  the  kind  of  material  and  kind  of  stress. 
It  simply  expresses  the  ratio  between  the  rates  of  stress  and 
strain. 

The  characteristic  strain  of  a  tensile  stress  is  evidently 
an  increase  of  the  linear  dimensions  of  the  body  in  the 
direction  of  action  of  the  external  forces. 

Let  this  increase  per  unit  of  length  be  represented  by 
/,  while  p  and  E  represent,  respectively,  the  correspond- 
ing intensity  and  coefficient.  Eq.  (i)  of  the  preceding 
article  then  becomes 

p=El,     or    £=| (i) 

E  is  then  the  coefficient  of  elasticity  for  tension. 

The  characteristic  strain  for  a  compressive  stress  is 
evidently  a  decrease  in  the  linear  dimensions  of  the  body 
in  the  direction  of  action  of  the  external  forces.  Let  Zt 
represent  this  decrease  per  unit  of  length,  pl  the  intensity 
of  compressive  stress,  and  El  the  corresponding  coefficient. 
Hence 

A=£A>     or    E,  =  ^ (2) 

l\ 

Ev  consequently  is  the  coefficient  of  elasticity  for 
compression. 


Art.   2.] 


COEFFICIENTS  OF  ELASTICITY. 


The  characteristic  strain  for  a  shearing  stress  may  be 
determined  by  considering  the  effect  which  it  produces 
on  the  layers  of  the  body  parallel  to  its  plane  of  action. 

In  Fig.  i  let  A  BCD  represent  one  face  of  a  cube,  another 
of  whose  faces  is  fixed  along  AD.  If  a  shear  acts  in  the 
face  EC,  whose  plane  is  normal  to  the  plane 
of  the  paper,  all  layers  of  the  cube  parallel 
to  the  plane  of  the  shearing  stress,  i.e.,  BC, 
will  slide  over  each  other,  so  that  the  faces 
AB  and  DC  will  take  the  positions  AE  and 
DF.  The  amount  of  distortion  or  strain 
per  unit  of  length  will  be  represented  by 
the  angle  EAB  =  </>.  If  the  strain  is  small, 
there  may  be  written  <j>,  sin  <£,  or  tan  <£ 
indifferently. 

Representing,  therefore,  the  intensity  of  shear,  coeffi- 
cient, and  strain  by  5,  G,  and  </>,  respectively,  eq.  (i)  of 
Art.  i  becomes 


FIG. 


S=G<f)t     or 


(3) 


It  will  be  seen  hereafter  that  there  are  certain  limits 
of  stress  within  which  eqs.  (i),  (2),  and  (3)  are  essentially 
true,  but  beyond  which  they  do  not  hold;  this  limit  is 
called  the  limit  of  elasticity,  and  is  not  in  general  a  well- 
defined  point. 

The  line  Okghn  exhibited  in  Fig.  2  represents  the  actual 
strains  in  a  piece  of  structural  steel  i  inch  in  length  with 
i  square  inch  of  cross-section.  0  is  the  origin  of  coordi- 
nates, and  the  loads  per  square  inch,  i.e.,  intensities  of 
stresses,  are  shown  by  the  vertical  ordinates  drawn  parallel 
to  OC  from  OD  to  the  strain  curve,  while  the  strains  per 
unit  of  length,  that  is,  per  inch,  are  laid  off  as  horizontal 
ordinates  of  the  curve  parallel  to  OD.  If  Op'  is  the  in- 


6  ELASTICITY  IN   AMORPHOUS  SOLID  BODIES.         [Ch.  I. 

tensity  of  stress,  pr  corresponding  to  the  point  k  of  the  strain 
curve,  while  01'  is  the  resulting  strain  per  unit  of  length, 
then  p'  =Elf .  Again,  if  g  is  at  the  upper  limit  of  the  straight 
portion  of  the  curve  for  which  the  intensity  of  stress  and 
rate  of  strain  are  p  and  /  respectively,  the  relation  between 
those  two  quantities  is  shown  by  eq.  (i).  Since  E,  also  as 


o 


b 


FIG.   2. 


shown  by  eq.  (i),  is  equal  to  the  quotient  of  p  divided 
by  /,  Fig.  2  shows  that  it  is  equal  to  the  tangent  of  the 
angle  between  OD  and  the  straight  portion  Og  of  the  strain 
curve,  it  being  supposed  that  the  rates  of  strain  are  laid 
down  at  their  actual  or  natural  sizes.  If  the  strain  line  is 
curved,  the  first  term  of  eq.  (2)  of  Art.  i,  the  differential 
ratio,  will  represent  the  tangent  of  the  angle  between  the 
curve  and  the  horizontal  axis  OD  in  Fig.  2.  The  point  g, 
being  at  the  upper  limit  of  constant  proportionality  be- 
tween intensity  of  stress  and  rate  of  strain,  is  called  the 
elastic  limit,  above  which  it  is  seen  that  the  strains  in- 
crease far  more  rapidly  than  the  stresses  until  the  point  n 
is  reached,  where  actual  rupture  takes  place.  The  nearly 
horizontal  portion  of  the  curve  between  g  and  h  and  a  little 


Art.  3.]          STRESSES   OF   TENSION  AND  COMPRESSION.  7 

above  g  indicates  the  "  yield  point,"  an  intensity  of 
stress  where  the  material  is  said  first  to  "break  down"  or 
stretch  rapidly  under  tensile  stress  without  much  increase 
of  the  latter. 


Art.  3. — Direct  Stresses  of  Tension  and  Compression. 

The  direct  stresses  of  tension  and  compression  always 
produce  shearing  stresses  and  strains  on  all  planes  in  the 
interior  of  a  body  except  those  perpendicular  and  parallel 
to  those  direct  stresses.  If,  in  Fig.  i,  a  straight  piece  of 
material  CD  is  subjected  to  the  tensile  stress  induced  by 
the  forces  P  equal  and  opposite  to  each  other,  there  will  be 
pure  tension  only  on  all  planes  or  sections  of  the  piece  at 
right  angles  to  the  direction  of  the  forces  P,  such  as  HK. 
On  all  planes  passing  through  the  longitudinal  axis  of  the 
piece  there  will  be  no  stress  whatever,  if,  as  is  supposed, 
the  forces  P  are  uniformly  distributed  over  the  sections 
of  application  DF  and  BC. 


H' 


K          K' 

FIG.  i. 

On  every  oblique  plane  or  section  in  all  parts  of  the 
piece  as  H'K1 ',  supposed  to  be  perpendicular  to  the  plane 
of  the  diagram,  there  will  be  shear  as  well  as  direct 
stress  of  tension  normal  to  it,  the  intensities  of  both  the 
shear  and  the  normal  stress  being  dependent  upon  the 
angle  a  between  HK  and  H'K'.  The  force  P  may  be 
resolved  by  the  triangle  of  forces  into  two  components, 
one  at  right  angles  to  H'K',  represented  by  TV,  and  the 
other  along  or  tangential  to  H'K',  represented  by  5.  If 


8  ELASTICITY  IN  AMORPHOUS  SOLID  BODIES.        [Ch.  I. 

A  represents  the  area  of  the  normal  section  HK,  the  area 
of  the  oblique  section  H'K'  will  be  A  sec  a.  The  value 
of  the  normal  stress  N  will  be  TV  =P  cos  a,  but  5  =  P  sin  a. 
The  intensity  of  the  normal  tensile  stress  on  H'K'  will  be, 
therefore, 

N  P  cos  a 


The  intensity  of  shear  on  the  same  plane  H'K'  will  be 

S          P  sin  a 

s=-ji  -  =-A  —    -=P  sin  a  cos  a.       .     .     (2) 
A  sec  a     A  sec  a 

\Vnen  the  angle  a  is  zero,  5  in  eq.  (2)  becomes  zero, 
while  n  in  eq.  (i)  becomes  equal  to  /?,  i.e.,  the  intensity  of 
direct  tensile  stress  on  the  normal  section.  On  the  other 
hand,  when  the  angle  a  has  the  value  of  90°,  both  n  and  s 
become  zero,  i.e.,  there  is  no  stress  whatever  on  a  longi- 
tudinal, axial  plane. 

Inasmuch  as  the  angle  a:  may  have  any  value  what- 
ever from  zero  to  90°  on  either  side  of  HK,  it  is  clear  that 
both  shearing  and  normal  tensile  stresses  will  be  found 
concurrently  on  every  oblique  plane  in  the  piece.  As  has 
been  observed  in  the  preceding  article,  these  shearing 
stresses  induce  the  lateral  strains  under  which  the  normal 
cross-sections  of  a  piece  subjected  to  pure  tension  decrease 
in  area  while  they  increase  under  the  action  of  pure  com- 
pression. 

Eqs.  (i)  and  (2)  have  been  written  on  the  assump- 
tion that  the  external  forces  P  produce  tension  in  the 
material,  but  precisely  the  same  equations  apply  to  the 
condition  of  pure  compression,  the  only  difference  being 
that  in  the  latter  case  the  external  forces  P  would  be  di- 
rected toward  each  other  from  the  ends  of  the  piece,  in- 
stead of  away  from  each  other. 


Art.  4.]  LATERAL  STRAINS. 


Art.  4. — Lateral  Strains. 

If  a  body,  as  indicated  in  Fig.  i ,  be  subjected  to  ten- 
sion, it  has  been  shown  in  Art.  3  that  all  of  its  oblique  cross- 
sections,  such  as  FE  and  GH,  will  sustain  shearing  stresses 
in  consequence  of  the  component  of  the  tension  tangential 
to  those  oblique  sections.  These  tangential  stresses  will 
cause  the  oblique  sections,  in  both  directions,  to  slide  over 

AGE  C 


B  F  H  D 

FIG.   i. 

each  other.  Consequently  the  normal  cross-sections  of  the 
body  will  be  decreased;  and  if  the  normal  cross-sections  of 
the  body  are  made  less,  its  capacity  to  resist  the  external 
forces  acting  on  A  B  and  CD  will  be  correspondingly  dimin- 
ished. 

If  the  body  is  subjected  to  compression,  oblique  sec- 
tions of  the  body  will  be  subjected  to  shears,  but  in  direc- 
tions opposite  to  those  existing  in  the  previous  case.  The 
effect  of  such  shears  will  be  an  increase  of  the  lateral 
dimensions  of  the  body  and  a  corresponding  increase  in 
its  capacity  of  resistance. 

These  changes  in  the  lateral  dimensions  of  the  body  are 
termed  "lateral  strains";  they  always  accompany  direct 
strains  of  tension  and  compression. 

It  is  to  be  observed  that  lateral  strains  decrease  a  body 's 
resistance  to  tension,  but  increase  its  resistance  to  com- 
pression. Also,  that  if  they  are  prevented,  both  kinds  of 
resistance  are  increased. 

Consider  a  cube,  each  of  whose  edges  is  a,  in  a  body 
subjected  to  tension.  Let  r  represent  the  ratio  between 


io  ELASTICITY  IN  AMORPHOUS  SOLID  BODIES.  [Ch.  I. 

the  lateral  and  direct  strains,*  and  let  it  be  supposed  to 
be  the  same  in  all  directions.  If  /,  as  in  Art.  2,  represents 
the  direct  unit  strain,  the  edges  of  the  cube  will  become,  by 
the  tension,  a(i+/),  a(i  —  Ir),  and  a(i—rl).  Consequently 
the  volume  of  the  resulting  parallelepiped  will  be 

a3(i+/)(i-r/)2=a3[i+/(i-2r)]      .     ...     (i) 

if  powers  of  /  higher  than  the  first  be  omitted.  With  r  be- 
tween o  and  i,  there  will  be  an  increase  of  vo!:une,  but  not 
otherwise. 

If  the  body  is  subjected  to  compression,  the  edges  of 
the  cube  become  a(i— /J,  a(i-fr1/1),  and  a(i+r1/1);  while 
the  volume  of  the  parallelepiped  takes  the  value 

a3(i  -/x)(i  +  r1/1)' -a«[i +/,("!- 1)].      .     .     (2) 

As  before,  the  higher  powers  of  ^  are  omitted.  If  the 
volume  of  the  cube  is  decreased,  rl  must  be  found  between 
o  and  J. 

If  a  be  unity  in  eq.  (i),  it  is  then  clear  that  the  expres- 
sion l(i  —  2r)  is  the  change  of  volume  of  a  unit  cube,  i.e., 
it  is  the  rate  of  change  of  volume  when  the  intensity  of  stress 
is  p=El.  Hence  if  this  rate  of  change  of  volume  be  mul- 
tiplied by  a  definite  volume  V  the  result  will  be  the  total 
change  of  that  definite  volume  produced  by  the  uniform 
intensity  of  stress  p. 

If  the  intensity  of  stress  varies  from  point  to  point  the 
total  change  of  volume  will  become : 


Evidently  the  volume  V  must  be  expressed  in  the  same 
independent    variable,  or    variables,    as    p.     The    integral 
must  then  be  made  to  cover  the  desired  limits. 
*  Frequently  called  Poisson's  ratio. 


Art.  5.]     RELATION  BETWEEN  COEFFICIENTS   OF  ELASTICITY-      n 


Art.   5. — Relation    between  the  Coefficients  of  Elasticity  for 
Shearing  and  Direct  Stress  in  a  Homogeneous  Body. 

A  body  is  said  to  be  homogeneous  when  its  elasticity, 
of  a  given  kind,  is  the  same  in  all  directions. 

Let  Fig.  i  represent  a  body  subjected  to  tension  parallel 
to  CD.  That  oblique  section  on  which  the  shear  has  the 
A  E  B  greatest  intensity  will  make 

an  angle  of  45°  with  either  of 
those  faces  whose  traces  are 
CD  or  BD ;  for  if  a  is  the  angle 
which  any  oblique  section 
makes  with  BD,  P  the  total 
tension  on  BD,  and  A'  the 
area  of  the  latter  surface,  the  total  shear  on  any  section 
whose  area  is  A'  sec  a  will  be  P  sin  a.  Hence  the  intensity 
of  shear  is 

P  sin  a       P 

=  — ,  sin  a  cos  a (i) 


G 

FIG.   i. 


sec  a 


A 


The  second  member  of  eq.  (i)  evidently  has  its  greatest 
value  for  a  =45°.     Hence  if  the  tensile  intensity  on  BD  is 

P 
represented  by  —r,  =  p,  the  greatest  intensity  of  shear  will  be 


Then  by  eq.  (3)  of  Art.  2, 


(2) 


(3) 


In  Fig.  i  EK  and  KG  are  perpendicular  to  each  other, 
while  they  make  angles  of  45°  with  either  AB  or  CD.  After 
stress,  the  cube  EKGH  is  distorted  to  the  oblique  paral- 
lelopiped  E'KG'H'.  Consequently  EKGH  and  E'KG'H' 
correspond  to  ABCD  and  AEFD,  respectively,  of  Fig.  i, 


12  ELASTICITY  IN  AMORPHOUS  SOLID  BODIES.          [Ch.  I. 


Art.  2.     The  angular  difference  EKG  —  E'KG*  is  then  equal 
to  4>  ;  and  EKE'  =  GKG'  •=  -.     Also,  E'KF'  =  45°  -  ^. 

2  2 

Using,   then,   the   notation  of    the   preceding   articles, 
there  will  result,  nearly, 


.      .      (4) 

4  /  -"-   ~T~  * 

remembering  that 

F'K=FK(i+l),     and     E'F'  =  FK(i  -rl). 

From   a    trigonometrical    formula    there   is    obtained, 
very  nearly, 

tan  45°  — tan—     i  —  — 


(i\ 
45-1= 


tan45°+tan- 


From  eqs.  (4)  and  (5), 

Substituting   from    eq.   (3),  as  well     as    from   eq.    (i)    of 
Art.  2, 

E 


It  has  already  been  seen  in  the  preceding  article  that  r 
must  be  found  between  o  and  \,  consequently  ike  coefficient 
of  elasticity  for  shearing  lies  between  the  values  of  J  and  \  oj 
that  of  the  coefficient  of  elasticity  for  tension. 

This  result  is  approximately  verified  by  experiment. 

Since  precisely  the  same  form  of  result  is  obtained  by 
treating  compressive  stress,  instead  of  terisile,  there  will  be 
found,  by  equating  the  two  values  of  G, 

E          E               E      i+r 
i+r=  i+r/    °r     ~E  ~  i+r *.' 


Art.  6.]  SHEARING  STRESSES  AND  STRAINS.  13 

It  is  clear,  from  the  conditions  assumed  and  operations 
involved,  that  the  relations  shown  by  eqs.  (7)  and  (8)  can 
only  be  approximate. 

Art.  6. — Shearing  Stresses  and  Strains. 

In  the  preceding  Arts,  the  more  simple  and  ordinary 
relations  between  stress  and  strain  are  shown,  but  in  this 
and  following  Arts,  it  is  desirable  to  give  a  more  extended 
treatment. 

Materials  are  rarely  used  in  structures  and  machines 
under  conditions  in  which  the  stress  is  wholly  shear.  The 
usual  conditions  are  such  as  to  produce  shear  concurrently 
with  stresses  of  tension  and  compression.  Even  in  the  use 
of  rivets,  where  shearing  stress  acts  prominently,  tension 
and  compression  in  the  form  of  flexure  and  direct  com- 
pression are  concurrent.  Again  in  the  case  of  flexure  or 
the  bending  of  beams,  the  shearing  stress  is  sufficiently 
high  in  intensity  in  some  cases  to  produce  failure,  but 
concurrently  with  relatively  high  intensities  of  tension 
and  compression. 

Figs,  i  and  2  show  a  rectangular  parallelepiped  of 
material  of  depth  b  at  right  angles  to  the  plane  ABCD 
firmly  held  on  the  face  AD,  while  the  intensities  of  shear 
s  and  5'  act  on  the  faces  AB,  BC,  CD,  and  AD.  It  is 
supposed  that  no  other  stresses  act  upon  the  exterior 
faces  of  the  prism  of  material.  Let  the  prism  be  imagined 
to  be  divided  into  indefinitely  thin  vertical  slices  at  right 
angles  to  the  face  ABCD  when  in  its  original  position 
shown  by  AB'C'D.  Similarly  let  the  prism  be  imagined 
to  be  divided  into  indefinitely  thin  horizontal  slices  at 
right  angles  to  the  same  face. 

Before  considering  the  distortion  of  the  prism  due  to 
the  action  of  the  shearing  stresses  an  important  but  simple 
principle  must  be  established.  As  there  are  no  stresses 


ELASTICITY  IN  AMORPHOUS  SOLID  BODIES.        [Ch.  I. 


acting  upon  the  prism  except  the  opposite  pairs  of  shearing 
stresses  whose  intensities  are  5  and  sr  as  shown,  it  is  clear 
that  the  prism  must  be  held  in  equilibrium  by  the  two 
couples  acting  in  opposite  directions  whose  lever  arms 
are  AB'  and  AD.  Let  /  represent  the  length  AB'  of  the 


FIG.  i. 


FIG.  2. 


prism,  while  AD=d,  as  shown  in  Fig.  2.     Then  since  the 
prism  is  in  equilibrium  there  will  result  the  equation, 


s'bl.d=sbd.l 
.'.     s=sf. 


This  equation  shows  that  the  intensities  of  two  shears 
acting  on  planes  at  right  angles  to  each  other  and  parallel 
to  a  third  plane  at  right  angles  to  the  other  two  must  be 
equal.  Furthermore,  it  is  clear  from  Fig.  2  that  the  shears 
on  the  faces  of  the  prism  must  act  in  pairs  toward  two  of 
the  corners  of  the  prism  diagonally  opposite  each  other 
and  away  from  the  other  diagonally  opposite  pair  of  corners. 

The  rectangular  prism  of  Figs,  i  and  2  may  be  con- 
sidered indefinitely  small  under  ordinary  conditions  of 


Art.  6.]  SHEARING  STRESSES  AND  STRAINS.  15 

stress  in  structural  material  in  order  to  have  the  stress 
uniformly  distributed  on  the  four  faces.  Whatever  may 
be  the  condition  of  stress  at  any  point  in  the  interior  of  a 
piece  of  material,  the  stresses  acting  upon  the  four  faces 
of  the  rectangular  prism,  when  all  stress  is  parallel 
to  one  plane,  may  be  resolved  into  normal  and  tangential 
components.  The  normal  components  will  act  opposite 
to  each  other  producing  no  moments  about  any  point, 
but  the  tangential  components  will  produce  precisely 
the  moments  shown  in  Figs,  i  and  2.  The  equilibrium 
of  the  indefinitely  small  prism  invariably  requires  there- 
fore the  action  of  two  pairs  of  shears  of  equal  intensity, 
as  established  above. 

The  complete  distortion  of  the  rectangular  prism 
A  BCD  may  be  considered  as  produced  first  by  the  sliding 
over  each  other  of  the  indefinitely  thin  vertical  sections 
parallel  to  BC,  so  as  to  produce  the  oblique  prism  AB"C^Dt 
Fig.  i,  then  by  the  subsequent  sliding  over  each  other  of 
the  indefinitely  thin  horizontal  sections  parallel  to  DC, 
so  as  to  produce  the  oblique  prism  AB"C"D' .  This  last 
movement  of  the  horizontal  slices  will  bring  the  line  AD 
into  the  position  of  AD' ',  then  swinging  the  latter  line  about 
A  to  the  original  position  AD,  the  completely  distorted 
prism  will  take  the  form  A  BCD. 

B'B",  Fig.  i,  is  the  characteristic  shearing  strain  pro- 
duced by  the  vertical  shearing  stress  whose  intensity  is 
s  acting  in  the  planes  parallel  to  BC.  DD'  is  the  character- 
istic shearing  strain  produced  by  the  action  of  the  hori- 
zontal shearing  intensity  s'  in  sliding  the  thin  horizontal 
slices  over  each  other.  These  detrusive  movements  are 
so  small  that  B'B"  may  be  considered  at  right  angles  to 
AB  and  DD'  at  right  angles  to  AD.  The  total  detrusive 
strain  B'B  is  the  sum  of  B'B",  due  to  the  vertical  shear, 
and  B"B  due  to  the  horizontal  shear,  and  B'B"  =B"B, 


16  ELASTICITY  IN  AMORPHOUS  SOLID  BODIES.        [Cn.  I. 

if  AB'  =AD.     The  total  shearing  strain  per  unit  of  length 
of  AB  will  therefore  be, 

B'B  _^B'B"+B"B 

AB=          AB (2) 


This  is  the  expression  for  the  characteristic  resultant 
shearing  strain  and  it  is  seen  to  be  measured  at  right  angles 
to  the  original  face  AB',  i.e.,  it  is  a  small  arc  measurement 
in  radians.  It  is  important  to  remember  that  this  total 
detrusive  strain  due  to  shear  is  the  sum  of  two  equal 
effects,  one  of  horizontal  shear  and  the  other  of  vertical 
shear,  i.e.,  of  the  two  shears  on  planes  at  right  angles  to 
each  other. 

If  b  =  i  and  if  AB'C'D,  Fig.  2,  now  be  considered  square 
so  that  AB=BC,  then  will  the  tension  T  acting  perpen- 
dicular to  the  plane  BD  be  equal  to  the  sum  of  the  com- 
ponents of  the  shear  s=s'y  on  the  planes  BC  and  DC, 
normal  to  the  diagonal  plane  BD.  Since  the  angle  BCA 

is  45°  and  its  cosine  —7=,  the  following  equation  at  once 

V  2 

results : 

7=25  COS  45°=sV7.  ....       (3) 

Similarly  the  compression  on  the  diagonal  plane  AC  is: 
C=-sV2. (4) 

As  the  area  of  each  diagonal  plane  section  AC  and  BD 
is  V7,  the  intensity  of  the  tension  T  and  compression  C 
on  the  planes  AC  and  BD  respectively  will  be: 


1       -   _      °       _o  (t\ 

/——         /— — •>•  .      .     •      •      .      v^y 

V2  V2 


Art.  6.]  SHEARING  STRESSES   AND   STRAINS.  17 

Hence  it  is  seen  that  when  the  stress  it  any  point  is 
wholly  shear  on  two  planes  at  right  angles  to  each  other 
and  perpendicular  to  the  plane  to  which  the  shearing 
stress  is  parallel,  the  stress  on  two  planes  at  right  angles 
to  each  other  and  making  angles  of  45°  with  the  two 
planes  on  which  the  shears  act,  will  be  wholly  tension  on 
one  and  compression  on  the  other,  and  both  will  have  the 
same  intensity  as  the  two  shears. 

Inasmuch  as  the  prism  whose  section  is  shown  in  Fig. 
2  is  subjected  to  a  normal  stress  of  tension  in  the  direction 
of  AC  and  an  equal  normal  stress  of  compression  in  the 
direction  BD,  it  is  obvious  that  there  will  be  no  change 
in  volume  due  to  those  stresses,  since  the  change  in  inten- 
sity caused  by  one  stress  will  be  exactly  neutralized  by  the 
other.  Again  the  sliding  over  each  other  of  the  thin 
slices  of  the  material  will  not  change  its  density  or  volume, 
although  a  change  of  shape  is  produced.  Hence  it  is  to 
be  carefully  observed  that  shearing  stresses  produce  no 
change  of  volume,  but  change  of  shape  only. 

If  0  is  the  angle  B'AB=C'DC,  then  in  general,  the 
resultant  shearing  strain  B'B  =  CfC=ABf(j>  =ABf  sin  0 
=  AB'  tan  0,  since  the  angle  0  is  exceedingly  small.  If 
AB=BC  =  i,  B'B  =  <j>=sm  0=tan  0. 

In  Fig.  2  if  the  total  detrusive  strain  CC'  be  projected 
on  the  diagonal  AC  the  change  CC\  in  length  of  that 
diagonal  will  result.  As  the  angle  C'CC\  is  45°,  the 

change  of  length  CC\  will  be  -7=,  and  the  strain  per  unit 


of  length  of  the  diagonal  will  be, 

0i0 


It  is  clear  that  the  diagonal  BD  will  be  shortened  by 
the  same  amount.     Indeed  Eq.  6  shows  the  tensile  strain 


i8  ELASTICITY  IN  AMORPHOUS  SOLID  BODIES.         [Ch.  I. 

in  the  diagonal  AC,  while  the  same  value  with  a  minus 
sign  would  show  the  compressive  strain  for  the  diagonal 
BD.  If  the  diagonal  AC  were  subjected  to  the  tensile 

intensity  5  only  the  strain  per  unit  of  length  would  be  — . 

h, 

If  G  is  the  modulus  of  elasticity  for  shearing,  the 
intensity  of  shearing  stress  may  be  written, 

s=s'=G$. .-    (7) 

Inasmuch  as  the  total  detrusive  strain  <£  per  linear 
unit  is  the  sum  of  the  equal  effects  of  the  shears  on  the 
two  faces  of  the  prism,  it  would  be  more  rational  to  call 

-  the  detrusive  strain  per  linear  unit  for  the  shear  on  one 
2 

face  of  the  prism.  This  would  make  the  modulus  G 
of  elasticity  for  shearing  double  the  value  usually  employed, 
but  it  would  represent  accurately  the  rigidity  of  the  material, 
since  one  half  of  the  total  shearing  strain  </>,  Fig.  2,  is  pro- 
duced by  a  rotation  of  the  prism  as  a  whole.  In  other 
words  the  total  strain  is  the  sum  of  two  separate  but  equal 
strains.  This  doubling  of  the  value  of  G  would  obviously 
change  no  results  of  computation  for  practical  purposes 
since  the  strain  </>  would  be  halved.  It  is  interesting 
to  observe  in  connection  with  this  feature  of  the  matter 
that  the  shearing  rigidity  of  the  material  in  this  case,  would 
become  the  .same  as  the  apparent  rigidity  in  tension  or 
compression. 

Art.  7. — Relation  between  Moduli  of  Elasticity  and  Rate   of 
Change  of  Volume. 

The  preceding  analyses  yield  some  simple  relations 
between  the  moduli  of  elasticity  for  tension,  compression 


Art.  7.]         RELATION  BETWEEN  MODULI  OF  ELASTICITY.  19 

and  shearing  and  the  rate  of  change  of  volume  z;,*  i.e.,  the 
change  of  unit  volume  for  unit  intensity  of  stress. 

In  Fig.  2  of  the  preceding  Art.  CCf  shows  the  total  shear- 

ing  strain    </>,    and   the   elongation   or  strain    CC\(  =  —/= 

\     V  2 

of  the  diagonal  AC.  It  has  also  been  shown  that  the  inten- 
sity of  tension  on  BD  or  compression  on  AC  is  the  same 
as  the  -shear  5=5'.  Remembering  that  the  compression 

5  on  AC  will  produce  a  unit  positive  lateral  strain  r  — 

k, 

parallel  to  AC,  the  two  equal  values  of  the  unit  strain  of 
the  diagonal  AC  may  be  written, 


2        2G  E         E' 

Hence, 

C  — 
~ 


If  the  modulus  of  elasticity  for  compression,  Ei,  should 
be  different  from  that  for  tension  it-is  evident  that  the  third 
member  of  Eq.  i  would  be  required. 

If  the  value  of  r  is  J  or  J  then  will, 

G  =  %E     or       E  ......     .     (2) 

The  relation  between  v  and  E  can  readily  be  written 
by  considering  a  cube  (indefinitely  small  if  necessary) 
subjected  to  uniformly  distributed  tensile  stress  of  inten- 
sity p  normal  to  each  of  its  six  faces.  Each  edge  of  the 
cube,  assumed  to  be  of  unit  length,  will  be  lengthened  by 

P 
the  normal  stress  parallel  to  it  to  the  extent  fe  and  it 


*  The  reciprocal  of  what  is  sometimes  called  the  volume  or  bulk  modulus. 


20  ELASTICITY  IN  AMORPHOUS  SOLID  BODIES.          [Ch.  I. 

will  be  decreased  in  length  r  -p  by  each  of  the  two  normal 

rL 

stresses  p  acting  at  right  angles  to  it. 

The  resultant  change  in  length  of  each  edge  will  then  be, 


Hence  the  change  of  unit  volume  in  terms  of  the  unit 
rate  v.  will  be, 


(3) 


If  V  be  any  volume,  the  total  change  of  volume  will 
be  pvV. 

The  equation  preceding  Eq.  (3)  shows  that  the  unit 
rate  of  change  of  volume  v  is  simply  the  sum  of  the  three 

linear  rates  of  change  of  the  edges  of  the  cube,  since  —  —  — 

h, 

is  the  change  of  length  of  each  edge  of  the  cube  for  each 

unit  of  p,  i.e.,  pi1     2r)  is  the  change  in  length  of  each  such 

\    &    I 

edge  under  the  action  of  the  intensity  of  stress  p.  If  the 
intensity  of  stress  parallel  to  each  edge  of  the  cube  should 
be  different  from  the  others  the  preceding  analysis  shows 
that  the  rate  of  variation  of  volume  would  still  be  the  sum 
of  the  three  coordinate  linear  rates  of  variation. 
By  the  aid  of  Eq.  (i), 


Therefore  : 

~ 


6+2Gv 


Art.  8.]  ALL  STRESSES  PARALLEL  TO  ONE  PLANE. 

Finally,  placing  r  from  Eq.  (5)  in  E^q.  (3), 


+Gv* 


21 


(6) 


These  simple  relations  will  enable  the  various  moduli 
to  be  determined  with  the  least  possible  amount  of  experi- 
mental work. 

Art.  8.— All  Stresses  Parallel  to  One  Plane — Resultant  Stress 
on  any  Plane  Normal  to  the  Plane  of  Action  of  the  Stresses. 

In  Fig.  i  let  XOY  be  the  plane  parallel  to  which  all 
stresses  act.  Then  OX  and  OY  being  any  rectangular 
coordinate  directions,  consider  the  two  planes  OA  and  OB 


normal  to  each  other  and  at  right  angles  to  the  plane 
XOY  and  let  the  width  of  each  of  those  planes  at  right  angles 
to  XOY  be  unity. 

Again  let  it  be  supposed  that  the  normal  stress  on  the 
plane  AO  has  the  intensity  pv  and  that  the  intensity  of 
the  tangential  or  shearing  stress  on  the  same  plane  is  pvx- 
Similarly  let  it  be  supposed  that  the  intensity  of  the  normal 
stress  on  the  plane  OB  is  pxj  the  intensity  of  the  tangential 


22  ELASTICITY  IN  AMORPHOUS  SOLID  BODIES.          [Ch.  I. 

or  shearing  stress  being  pxy.  It  is  known  from  the  prin- 
ciples already  established  in  the  preceding  articles  that  the 
two  intensities  of  shear  pvx  and  pxv  are  equal  to  each  other. 
The  problem  is  to  determine  the  intensity  and  direction 
of  the  resultant  stress  on  any  plane  AB,  taken  at  right 
angles  to  XOY.  In  general  the  resultant  stress  CD  will 
make  the  angle  <f>  with  the  normal  CF  to  the  plane  AB, 
i.e.,  the  resultant  stress  will  have  the  obliquity  </>. 

The  direction  of  the  plane  A  B  will  be  fixed  by  the  angle 
which  its  normal  CF  makes  with  the  axis  OX.  In  order 
that  the  stresses  on  the  three  planes  in  question  may  be 
taken  as  uniformly  distributed  let  it  be  assumed  that 
OA  =d%  and  OB  =dy.  Then  will 

AB  =dy  sec  a.  =dx  cosec  a.      .     .     .     .     (i) 

If  p  is  the  intensity  of  the  uniformly  distributed  result- 
ant stress  on  AB,  then  the  equilibrium  of  the  indefinitely 
small  triangular  prism  OAB  requires  that  the  two  following 
equations,  representing  the  sums  of  all  the  forces  acting 
upon  it  in  the  two  coordinate  directions,  shall  be  true. 

pxdy+pxvdx=p  cos  («.+  </>).  dy  sec  a       .       (2) 
pydoc  -\-pxydy  =  p  sin  (a  +  0)  .  dy  sec  a       .       (3) 

Fig.  i  shows  that  dytan.a=dx.  Hence  Eqs.  (2)  and 
(3)  become  Eqs.  (4)  and  (5),  respectively: 


px  cot  a.+pxv=p  cos  (<*  +  </>)  cosec  a        .     .     (4) 

Pv+Pxv  COt  a  =  p  sin  («  +  <£)  COS6C  a.          .      .       (5) 


It  is  sometimes  convenient  to  express  the  normal  and 
tangential  components  of  the  resultant  intensity  p  in  terms 
of  the  known  intensities  px,  pv  and  pxy.  If  in  Fig.  i  the 
stresses  on  the  faces  OA  and  OB  be  resolved  into  compo- 


Art.  8.]  ALL  STRESSES  PARALLEL   TO  ONE  PLANE.  23 

nents  normal  and  parallel  to  the  plane  AB  the  sum  of  the 
normal  components  will  be  equal  to  the  normal  stress  on 
AB  while  the  sum  of  the  parallel  components  will  be  equal 
to  the  tangential  or  shearing  stress  on  A  B.  This  pro- 
cedure will  give, 

pydoo  sin  a+pyxdx  cos  a  -\-pxdy  cos  a-\-pxydy  sin  a 

=  pdy  sec  a  cos  <£. 

pydoc  cos  a—pyxdoc  sin  a—pxdy  sin  a+pxydy  cos  a 

=  pdx  cosec  a.     sin  </>. 

Using  the  values  already  given  for  dy  and  A  B  the  fol- 
lowing expressions  for  the  normal  and  tangential  compo- 
nents of  p  (p  cos  0  and  p  sin  </>)  will  result: 


py  sin2  a+px  cos2  a  -\-2pxV  sin  a  cos  a  =p  cos  0     . 
(pv—px)  sin  a  cos  o;+^(cos2  a—  sin2  a)  =£  sin  0.     (5a) 


These  two  equations  will  be  used  in  establishing  the 
ellipse  of  stress  in  the  next  Art. 

If  the  stress  p  is  a  principal  stress  its  obliquity  0,  i.e., 

the   angle  between  its  direction  and  the  normal  to  the 

plane  on  which  it  acts,   will  be  zero.     If   0  =  o  Eqs.  (4) 
and  (5)  become, 

p-px=pxvtana,          ....  (6) 

p  —  pv  =  pxv  cot  a  .....     •     (7) 
Subtracting  Eq.  6  from  Eq.   7, 

" 


cot«-tan«= 


tan  2  a        pxv 


taxi  2<x  =-*-.  (8) 

P*-pv 


24  ELASTICITY  IN  AMORPHOUS  SOLID  BODIES  [Ch.  I. 

If  the  angle  a\  satisfies  this  equation,  then  will  0:1+90° 
also  satisfy  it.  Hence,  -there  will  always  be  two  prin- 
cipal planes  at  right  angles  to  each  other  on  each  of  which 
a  normal  stress  only  acts,  i.e.,  there  is  no  shearing  stress 
on  either  principal  plane. 

Eq.  8  will  at  once  locate,  by  the  two  values  of  a,  the 
two  principal  planes,  while  the  same  two  values  of  a  intro- 
duced into  either  Eq.  6  or  Eq.  7  will  give  the  two  intensi- 
ties of  principal  stresses  to  be  called  p\  and  p2,  it  being 
supposed  that  the  normal  and  shearing  stresses  on  the 
planes  OA  and  OB  are  completely  known. 

The  two  principal  stresses  can  however  readily  be 
found  without  computing  the  angle  a.  Multiplying  Eq. 
7  by  Eq.  6, 


The  solution  of  this  quadratic  equation  gives, 


The  two  roots  of  this  equation  will  give  the  two  prin- 
cipal intensities  at  any  point  in  terms  of  the  known  inten- 
sities pz,  py  and  pxy. 

The  two  stress  intensities  px  and  pv  have  been  taken 
of  the  same  kind,  tension  or  compression,  and  considered 
positive.  If  one,  as  pv,  be  considered  compression  or 
negative,  its  sign  would  be  changed  in  the  preceding 
equations,  but  there  would  be  no  other  change. 

Sum  of  Normal  Components. 

If  any  other  plane  be  taken  at  right  angles  to  XOY, 
Fig.  i,  and  at  right  angles  to  the  plane  whose  trace  is  AB, 
the  preceding  equations  are  made  applicable  to  it  by  writing 


Art.  8.]  ALL  STRESSES  PARALLEL  TO  ONE  PLANE.  25 

90°  -J-QJ  for  a  in  Eqs.   (40)  and  ($a),  since  the  new  plane 
is  at  right  angles  to  that  whose  trace  is  AB. 

Then  in  Eqs.  (40)  and  (5  a)  there  must  be  written, 

For  sin  a,  sin  (90+0;)     =cos  a. 

For  cos  a,  cos  (90+0:)    =  —sin  a. 

Hence  by  Eq.  (40),  writing  £'  and  </>'  for  p  and  0; 

£„  COS2  a+px  Sin2  a  —  2pxv  sin  a  COS  a  =  £>'  COS  </>'. 

Then  by  adding  this-  equation  to  Eq.  (40)  ; 

P*+Pv=P  cos  </>+£'  cos  0'.        ...     (10) 


This  equation  shows  that  on  any  two  planes  at  right 
angles  to  each  other  the  sum  of  the  normal  intensities  will 
be  constant  and  equal  to  px-\-p^.  Furthermore,  inasmuch 
as  there  is  no  shear  on  the  principal  planes,  i.e.,  the  stress 
is  wholly  normal,  it  is  thus  seen  that  the  sum  of  the  normal 
intensities  on  any  two  planes  at  right  angles  to  each 
other  is  always  equal  to  the  sum  of  the  two  principal 
intensities. 

If  the  above  values  of  sin  a  and  cos  a.  are  written  in 
Eq.  (50),  the  following  equation  will  result: 

(pv  —  px)  sin  a  cos  a  -\-p:ry(cos2  <*  —  sin2  a)  =  —  p'  sin  </>'. 

This  equation  is  identical  with  Eq.  (50),  except  that 
the  sign  of  the  second  member  is  changed.  This  result 
simply  shows  what  is  already  known  that  the  intensities 
of  the  shears  on  planes  at  right  angles  to  each  other  are 
equal.  The  change  of  sign  indicates  the  direction  only  of 
the  shear. 

In  all  the  usual  cases  of  stress  arising  in  the  subject  of 
Resistance  of  Materials-  the  internal  stresses  produced  by 


26  ELASTICITY  IN  AMORPHOUS  SOLID  BODIES.          [Ch.  I. 

external  loading  may  be  considered  parallel  to  one  plane. 
The  preceding  investigation  shows  that  without  considering 
the  elastic  properties  of  the  material  there  are  two  equations 
of  condition  (Eqs.  4  and  5)  from  which  the  two  rectangular 
components  of  the  resultant  stress  p  (or  intensity  p  and 
obliquity  </>)  may  be  found.  If  the  general  case  of  internal 
stress  be  taken  in  wrhich  stresses  may  act  in  three  rectangu- 
lar coordinate  directions,  there  obviously  will  be  three 
equations  of  condition  from  which  the  three  rectangular 
components  of  the  resultant  stress  on  any  plane  may  be 
found. 

The  triangle  OAB  may  be  considered  the  side  of  a, 
wedge  whose  edge  is  at  0.  The  two  faces  OB  and  OA  are 
acted  upon  by  the  stresses  indicated  and  their  resultant 
holds  in  equilibrium  the  stress  p  on  the  head  A  B  of  the 
wedge.  The  surface  A  B  may  be  considered  a  part  of  the 
exterior  surface  of  a  body  acted  upon  by  the  stress  whose 
intensity  is  p,  while  the  faces  OA  and  OB  are  interior 
surfaces  of  the  body  acted  upon  by  the  internal  stresses 
shown. 

Art.  9. — The  Ellipse  of  Stress — Greatest  Intensity  of  Shearing 
Stress— Equivalence  of  Pure  Shear  to  Two  Principal  Stresses 
of  Opposite  Kinds  but  Equal  Intensities — Greatest  Obliquity 
of  Resultant  Stress  on  any  Plane. 

The  analysis  of  the  preceding  article  makes  it  compara- 
tively easy  to  express  the  relation  between  the  stress  on 
any  plane  whatever  at  right  angles  to  the  plane  parallel 
to  which  the  principal  stresses  act  and  those  principal 
stresses,  all  stresses  still  acting  parallel  to  one  plane.  In 
Fig.  i  let  OX  and  OY  be  taken  in  the  direction  of  the 
principal  stresses,  OA  representing  the  intensity  pi  of  the 
principal  stress  at  0  on  the  plane  ODt  while  OB  represents 


Art.  9.]  THE  ELLIPSE   OF  STRESS.  27 

the  intensity  of  the  principal  stress  p2,  acting  at  O  on  the 
plane  OC.  OCD  represents  an  indefinitely  small  triangular 
prism  whose  face  CD  normal  to  XOY  makes  the  angle  0 
with  the  principal  plane  OD.  The  intensity  of  the  re- 
sultant stress  on  any  plane  CD  is  represented  by  p,  whose 
obliquity  is  </>,  the  normal  TV  to  the  plane  CD  making  the 
angle  /3  with  the  axis  OX.  The  resultant  intensity  p  may 
at  once  be  written  by  the  aid  of  Eqs.  4  and  5  or  ^a  and 
5  a  of  the  preceding  article  if  the  principal  intensities  pi 
and  p2  be  written  in  the  place  of  px  and  pv,  respectively, 
in  those  equations  while  pxv  is  made  equal  to  zero.  This 
procedure  with  Eqs.  (40)  and  (50)  will  give  the  following 
Eqs.  (i)  and  (2). 


p2  sin2  0+£i  cos2  (3=p  cos  0,         .     .     .      (i) 
(p2-pi)  sin  ]8  cos  )8  =  £in£l  s{n  2/3  =p  sin  0.     .      (2) 

2 

Squaring  each  of  those  equations  and  adding  the  results  : 

/3=£2.         .     .     .     (3)* 


This  is  the  equation  of  an  ellipse  with  the  origin  of 
coordinates  at  the  centre,  the  rectangular  coordinates  being 
p2  sin  |8  and  pi  cos  0.  Fig.  i  shows  the  ellipse  of  stress, 
the  intensities  of  the  principal  stresses  being  represented 
by  the  semi  axes  of  the  ellipse. 

OB=p2     and     OA=pi. 

In  this  Fig.  pz  represents  the  intensity  of  the  principal 
stress  on  the  plane  OC,  while  pi  is  the  intensity  on  the 
principal  plane  OD.  The  intensity  p  on  any  plane  as  CD 

*  Precisely  the  same  result  will  be  obtained  by  making  pzv=oineqs. 
4  and  5  of  the  preceding  Art.  and  then  squaring  and  adding  them. 


28  ELASTICITY  OF  AMORPHOUS  SOLID  BODIES.  [Ch.  I. 

perpendicular  to  XOY  and  whose  normal  ON  makes  the 
angle  /3  with  OX  is  represented  in  Fig.  2  by  OH,  the  curve 
AHB  being  an  ellipse.  Let  the  partial  circles  shown  be 
described  by  the  radii  OB  and  OA.  Then  if  OCD  be  con- 
sidered indefinitely  small  the  normal  ON,  and  the  line 
OH  representing  the  intensity  of  the  resultant  stress  on 
the  plane  CD,  will  both  pass  through  the  origin  O.  Then 
OG  will  represent  p2  and  OK=p2  sin  0.  The  same  con- 
struction shows  that  HK—p\  cos  (3  since  OJ=p\.  The 
square  of  OH  =  p  will  then  obviously  be  equal  to  the  square 
of  HK  added  to  the  square  of  OK,  an  expression  identical 
with  Eq.  (3). 

Any  radius  vector  of  the  ellipse  therefore  represents 
the  intensity  of  a  resultant  stress  on  a  plane  whose  normal 
makes  the  angle  0  with  the  axis  of  X.  The  obliquity  of 
the  resultant  stress  in  question  is  represented  by  the  angle  0. 

The  two  principal  stresses  have  been  taken  of  the  same 
kind  in  finding  the  ellipse  of  stress,  but  the  results  are  es- 
sentially the  same  if  the  principal  stresses  are  of  opposite 
kind.  If  for  example,  p2  were  negative  while  pi  remains 
positive  p2  =  OB  would  be  laid  off  in  Fig.  i  to  the  left 
of  0  instead  of  laying  it  off  to  the  right  of  the  same  point. 
Similarly  if  the  sign  of  pi.  should  be  considered  negative 
that  intensity  would  be  laid  off  downward  from  0  to  A' 
instead  of  upward  to  A. 

If  the  two  intensities  of  principal  stresses  p\  and  p2 
are  equal  to  each  other  and  of  the  same  kind  Eq.  3  becomes 
pi  =  p2=p. 

Under  the  same  conditions  Eq.  (2)  shows  that  the 
shearing  intensity  is  zero,  whatever  value  the  angle  /3 
may  have,  since  in  such  a  case  pi—p2=o.  Hence  all 
stresses  are  principal  stresses  and  of  equal  intensity.  This 
condition  of  stress  is  the  same  as  that  which  holds  in  a 
perfect  fluid. 


Art.  9  J 


THE  ELLIPSE  OF  STRESS. 


29 


An  examination  of  the  ellipse  of  stress  as  given  in 
Fig.  i  shows  that  the  intensity  of  one  principal  stress  is 
greater  than  that  of  any  other  stress  at  the  point  for  which 
the  ellipse  is  drawn,  while  the  intensity  of  the  other  prin- 
cipal stress  is  the  least  of  all  the  intensities  at  the  same 
point,  since  the  semi-major  and  semi-minor  axes  of  the 
ellipse  are  the  greatest  and  least,  respectively,  of  all  -the 
semi- diameters.  If  therefore  in  the  design  or  construc- 
tion of  any  machine  or  structure  the  principal  stress  at 


any  point  is  provided  for  by  the  use  of  a  proper  working 
stress,  no  further  provision  for  the  direct  stresses  of  ten- 
sion and  compression  will  be  needed.  If  there  may  be 
either  a  reversal  of  stress  or  rapid  repetition  of  stresses  the 
intensity  of  working  stress  must  ^e  prescribed  under  a 
proper  recognition  of  those  conditions.  Similarly  provi- 
sion must  be  made  for  the  greatest  shearing  stress  at  the 
point  under  consideration. 

Greatest  Intensity  of  Shearing  Stress. 

The  intensity  of  shear  on  any  plane  CD  at  the  point 
0,  Fig.  i,  is  p  sin  0  as  given  by  Eq.  2.     Its  greatest  value 


30  ELASTICITY  IN  AMORPHOUS  SOLID  BODIES.  [Ch.  I. 

and  the  plane  on  which  it  acts  are  readily  determined  by 
differentiating  that  equation: 


cos  2(3. 


Hence, 

cos/3=sinj3;    or,  18=45°.     .....     (4) 

As    sin    45°  =cos  45°  =*—  /=,    the    greatest    intensity    of 

V  2 

shear  at  any  point,  as  O,  Fig.   i,  is  found  by  substituting 
/3=45°  in  the  second  member  of  Eq.  (2): 


The  planes  of  greatest  shear,  therefore,  are  at  the  angle 
°f  45°  from  each  of  the  two  principal  planes,  and  the  greatest 
intensity  of  shear  is  half  the  difference  of  the  principal 
intensities,  both  of  the  latter  being  of  the  same  kind. 

As  j8=45°  the  resultant  intensity  of  stress  on  the  plane 
of  greatest  shear  will  be,  by  Eq.  (3), 


If  p2  =  ±pi  ;  p  =  ±p2  =  ±pi. 


(so) 


Equivalence  of  Pure  Shear  to  Two  Principal  Stresses  of 
Opposite  Kinds  but  Equal  Intensities. 

If  the  principal  stresses  are  of  opposite  kinds,  i.e.,  if 
pi  is  negative  while  p2  is  positive,  then  by  Eq.  (5)  the 
greatest  shear  becomes  : 

.     .     (6) 


Art.  9.]  EQUIVALENCE  OF   PURE  SHEAR.  31 

The  greatest  intensity  of  shear  is  half  the  sum  of  the  prin- 
cipal intensities. 

Obviously  the  planes  of  greatest  shear  remain  as  estab- 
lished by  Eq.  (4). 

If  the  principal  stresses  of  opposite  kinds  have  the  same 
intensities  Eq.  (6)  shows  that : 

pt=p2=pi (7) 

Hence  the  intensity  of  the  greatest  shear  is  the  same 
as  that  of  the  principal  stresses  of  opposite  kinds.  It  is 
therefore  sometimes  stated  that  a  pair  of  normal  stresses 
of  opposite  kinds  and  equal  intensities  on  two  planes  at 
right  angles  to  each  other  are  equivalent  to  two  pure 
shears  of  the  same  intensity  as  the  normal  stresses  on 
planes  at  right  angles  to,  each  other,  but  at  45°  with  the 
planes  on  which  the  normal  stresses  act,  all  planes  under 
consideration  being  perpendicular  to  one  plane.  This 
simple  condition  of  stress  exists  in  both  flexure  or  bending 
and  torsion,  and  some  important  results  are  based  on  it. 


Greatest  Obliquity  of  Resultant  Stress  on  any  Plane. 
If  Eq.  (2)  be  divided  by  Eq.  i : 


It  is  desired  to  find  that  value  of  /3  which  will  make  <£ 
(or  tan   0)   a  maximum.     By  differentiating  Eq.    (8)   and 

placing  -^-a7n   •    =o,  there  will  result, 

d(3 

(cos2  0  -sin2  j8)(/?2  sin2  /3-f£i  cos2  0) 

=  2  sin2  0  cos2  0(^2  -£i).  .     .     .     (9) 


32  ELASTICITY  IN  AMORPHOUS  SOLID  BODIES.        [Ch.  I. 

Remembering   that    cos2   /3  —  sin2   /3=cos    2/3   and  that 
2  sin  /3  cos  /3=sin  20,  Eq.  (9)  will  become  Eq.  (10); 

cos  2p(p2  sin2  0+pi  cos2  /?)  =sin  s/3  sin  0  cos  $(p2—p\).    (10) 


Calling  the  normal  component  of  the  intensity  p,  i.e., 
p  cos  <f>=pn  and  the  tangential  or  shearing  component 
p  sin  <f>=pt,  those  values  taken  from  Eqs.  (i)  and  (2) 
placed  in  Eq.  (10)  will  give, 

cos  2(3pn  =sin  2(3pt. 

Hence,  tan  2$=—  =  cot  </>=  tan  (90°  -</>).      .     .     (n) 

Pt 

And,  j8=45°-|.      ......     (12) 

Eq.  (12)  gives  the  relation  between  /5  and  0  when  the 
obliquity  0  is  the  greatest  possible. 
By  the  aid  of  Eq.  (12), 

sin  /3  cos  )8  =  |  sin  2/3  =  J  cos  0. 

Then  as,  sin2(45°—  -  )  =-(i  —sin  0), 

•    \  2/2 

and  cos2/  45°—  -)  =-(i  +sin  0), 

V  2/2 

Eq.  (8)  gives, 


cos  0  p2(i  —sin  0)  -\-pi(i  +sin 

Hence, 

Pi     i  —sin 


-         — 
i+sm  0 


,     and,     sm0=^—  ^.  .     .      (12) 

v  ^; 


Art.  10.]  ELLIPSE  OF  STRESS.  33 

The  relation  shown  in  the  first  of  Eqs.  (13)  is  used  in 
the  theory  of  earth  pressure.  The  second  of  Eqs.  (13) 
gives  the  value  of  the  greatest  obliquity  </>  in  terms  of  the 
known  principal  intensities  pi  and  p2. 

The  angle  0  locating  the  plane  on  which  the  obliquity 
is  greatest  may  also  be  expressed  in  terms  of  pi  and  p2. 

Using  Eqs.   (12)  and  (13), 

pi  _i  —sin  0  _i  —cos  20  _  sin2  ft 
p2     i  +sin  0     i  -f  cos  2/3     cos2  ft' 


The  intensity,  pr,  of  this  stress  of  greatest  obliquity 

is,    by   Eq.    (3),    since   by   Eq.    (14)    sin2    0=  —  —  —  and 

i 


ds) 


Art.  10. — Ellipse  of  Stress  and  Resulting  Formulae  for  the  Special 
Case  of  Zero  Intensity  of  One  of  the  Known  Direct  Stresses. 

If  in  the  second  preceding  article  it  be  supposed  that 
the  intensity  of  one  of  the  direct  stresses  as  px  is  zero 
while  the  other  intensity  py  and  the  two  shearing  inten- 
sities pxy=pvx  have  known  values,  the  formulae  will  be 
correspondingly  simplified.  This  is  the  condition  of  stress 
in  a  bent  beam  as  will  be  seen  later  on.  The  intensity 
of  direct  stress  pv  is  what  is  ordinarily  called  the  fibre 
stress  at  any  point  in  the  beam  and  this  intensity  varies 
directly  as  the  distance  from  an  intermediate  plane  (before 
flexure)  in  the  beam  called  the  neutral  surface.  The 
plane  OY  of  Fig.  i,  representing  part  of  a  beam,  is  su'p- 


34 


ELASTICITY  IN  AMORPHOUS  SOLID  BODIES.        [Ch.  I. 


posed  to  be  a  horizontal  plane  coincident  with  or  parallel 
to  the  neutral  surface  of  the  bent  beam  at  any  point,  while 
the  plane  whose  trace  is  OX  is  the  plane  of  vertical  (normal) 
transverse  section  of  the  beam  at  any  point.  Both  the 
direct  intensity  pv  and  the  intensity  of  shear  pxv  are 
readily  determined  from  the  known  conditions  of  loading 
and  flexure.  The  analysis  of  this  condition  of  stress 
therefore  is  of  much  practical  importance  in  connection 
with  the  design  or  other  treatment  of  beams  subjected  to 
transverse  bending. 


FIG.  i. 


If  the  stress  p  in  Eq.  (40)  of  Art.  8  is  a  principal 
stress  and  if  the  intensity  px  =  o,  the  principal  intensity 
p  will  become, 

p  =pv  sin2  a  +  2pxy  sin  a  cos  a.        ...     (i) 


Or,  Eq.  (9)  of  the  same  Art.  will  give  for  the  two  prin- 
cipal intensities, 

(2) 


Also  Eq.  (8)  of  the  same  Art.  will  give, 

tan  2a  =  -&S-.  (3) 


Art.  10.]  ELLIPSE  OF  STRESS.  35 

If  the  point  0,  Fig.   i,  is  in  the  neutral  surf  ace  of  the 
bent  beam  pv=o;  and,  hence, 

tan  2a  =  —  oo ,     or,     2^  =  ^90°.      ...     (4) 
Therefore,  a  =  =±145°. 


If  the  stress  pv  is  negative  or  compression,  a  =  \  Q. 

The  direct  fiber  stresses  in  a  bent  beam  are  tensile  on  one 
side  of  the  neutral  surface  and  compressive  on  the  other. 

As  in  this  special  case  a  =—45°,   sin  «=—  cos  a=  --  ^= 

V2 

and  the  intensity  p  of  the  principal  stress  becomes  by  the 
aid  of  Eq.  (i),  since  pv  =o. 

P=-p*v  .....     •     •     (5) 

It  has  already  been  seen  that  a  and  90°  +«  will  satisfy 
Eq.     (3);      but     90°  +a  =90°—  45°  =45°.     Hence    placing 
inEq.  (i), 


Therefore  at  0,  Fig.  i,  where  there  is  no  direct  stress  (but 
shear  only)  on  the  two  planes  OX  and  OY  the  principal 
stresses  are  of  equal  intensities,  but  of  opposite  kinds  and 
they  act  on  planes  making  angles  of  45°  with  the  planes 
OX  and  OY.  This  is  the  same  condition  shown  by  Eq. 
(7)  of  the  preceding  Art. 

Again  at  the  exterior  surface  of  the  beam  p  has  its 
greatest  value  and  the  shearing  intensity  pvx  =  o.  Eq.  (2) 
then  gives, 

20.  =o     or     180°. 

Hence,  «=o    -or     90°  ......     (6) 


36  ELASTICITY  IN  AMORPHOUS  SOLID  BODIES.  [Ch.  I- 

Eq.  i  gives  p  =  o  for  the  principal  stress  corresponding  to 
«=o;  and,  for  a  =90°, 


(7) 


There  is  therefore  only  one  principal  stress  pv,  the 
fiber  stress  acting  on  the  normal  section  of  the  beam  for 
which  a  =90°. 

For  intermediate  points  of  the  beam  between  the 
neutral  surface  and  the  exterior  surface  the  principal 
stresses  will  have  varying  values  between  pxy  and  pv  as 
shown  by  Eqs.  (5)  and  (7)  with  planes  of  action  located 
by  values  of  a  between  ±45°  and  +90°. 

A  graphical  representation  of  this  condition  of  stress 
for  a  bent  beam  may  be  found  in  Art.  34- 

Art.  ii.  —  General  Condition  of  Stress  —  Ellipsoid  of  Stress. 

The  conditions  of  stress  in  structural  material  as  found 
within  the  experience  of  engineers  seldom  include  more 
than  the  action  of  stresses  parallel  to  one  plane.  There 
may,  however,  be  occasional  cases  in  which  an  elementary 
consideration  of  stresses  acting  in  any  direction  whatever 
becomes  necessary  or  at  least  helpful.  In  this  Article 
therefore  only  the  most  elementary  results  of  the  action 
of  such  stresses  will  be  treated,  including  the  ellipsoid 
of  stress. 

In  the  preceding  articles  both  the  determination  and 
the  application  to  a  number  of  useful  cases  of  the  ellipse 
of  stress  have  been  made.  That  ellipse  is  simply  a  special 
case  of  the  more  general  ellipsoid  of  stress.  In  other 
words,  if  the  action  of  stresses  in  space,  i.e.,  on  three 
rectangular  coordinate  planes  be  considered  it  will  be  found 
that  there  will  be  three  such  planes  at  any  point  on  which 
there  will  be  no  shear  and  which  therefore  are  called  prin- 


Art.  ii.] 


GENERAL  CONDITION  OF  STRESS. 


37 


cipal  planes,  the  resultant  normal  stresses  being  called 
the  principal  stresses  at  that  point.  The  semi-diameter 
of  the  ellipsoid  of  stress  drawn  with  its  center  at  the  point 
under  consideration  will  be  the  intensity  of  stress  in  that 
direction,  acting  upon  a  plane  whose  position  may  be 
determined.  For  this  elementary  treatment  let  the  three 
rectangular  coordinate  planes  in  Fig.  i  be  drawn. 


FIG.  i. 


In  that  Fig.  the  normal  stresses  on  the  planes  XOY, 
YOZ,  and  ZOX  have  the  intensities  pz,  px  and  py,  respect- 
ively. The  intensities  of  the  shearing  stresses  on  the 
planes  XOY  and  XOZ,  parallel  to  YOZ,  are  pzv=pvz', 
and  those  on  the  planes  XOY  and  YOZ,  parallel  to  ZOX, 
are  pzx=pxg',  and  finally  those  on  the  planes  YOZ,  and 
XOZ,  parallel  to  XOY,  are  pya,=pxy.  If  these  normal  and 
shearing  or  tangential  stresses  on  the  three  faces,  AOB, 
BOC,  and  AOC  of  the  elementary  tetrahedron  ABCO 


38  ELASTICITY  IN  AMORPHOUS  SOLID  BODIES.  [Ch.  I. 

are  given,  it  is  required  to  find  the  resultant  intensity  of 
stress  on  the  plane  surface  ABC*  the  base  of  the  tetra- 
hedron, and  its  obliquity.  It  may  be  considered  that 
AO  =dx,  B0=dy,  and  C0=dz.  It  will  simplify  the  result- 
ing equations  if  there  be  written  for  the  areas  of  the 
faces  of  the  elementary  tetrahedron; 


dxdy      ,     dydz  , 

a= -;     b=— — ;     and 


_dxdz 

C  —  • 


Also  if  area  ABC  —  J,  and  if  the  angles  which  the  normal 
TV  to  the  face  ABC  makes  with  axes  of  X,  Y,  and  Z, 
respectively,  are  a,  (3,  and  7-,  there  may  be  written: 

J  —a  sec  7  =  6  sec  a  =c  sec  /3.         .     .     .     (i) 

The  tetrahedron  is  held  in  equilibrium  by  the  normal 
and  shearing  stresses  on  the  faces  a,  b,  and  c  and  the 
resultant  stress  whose  intensity  is  q  on  ABC  =  A.  The 
components  of  that  resultant  parallel  to  the  axes  of  X, 
Y,  and  Z  whose  intensities  are  qx,  qv,  and  qz  are  respect- 
ively equal  and  opposite  to  the  corresponding  axial  sums 
of  stresses  as  shown  by  the  following  equations  : 


.....  (2) 
.....  (3) 
.....  (4) 

As  these  are  rectangular  components,  if  their  squares 
are  added  the  sum  will  be  equal  to  q2A2.  If  both  sides 
of  the  resulting  equation  be  divided  by  J2,  remembering 
that 

a2  b2  c2 

—-  =cos2  r\  -fi  =cos2  a;  —  =cos2  0; 


Art.  ii.]  GENERAL  CONDITION  OF  STRESS.  39 

ab  be  -,     ac 

—  =  cos«cosr;     -^  =  coso:cos|8;     and     --  =  cos  0  cos  r» 

and  that 

cos2  a+cos2  /3+cos2  7-  =  !;      ....     (5) 

there  will  be  found  : 


px2  COS2  a+py2  COS2  /3+£*2  COS2  ?-  +  2  COS  a  COS 

Pxypzy  -\-pzpxz)  +2  COS  a  COS  $(pxpyx+pxzpvz+Pvpxy)  + 
2  COS  ]8  COS  r(P*Pvz+Pzxpxy+PvPzu)  +  P2xz(l  "COS2  |8)  + 

)=g2     ......     (6) 


The   square   root  of   the  first  member  of  eq.    (6)   will 

give  the  desired  value  of  the  intensity  q  on  any  given  plane. 

If  both  members  of  eqs.  (2),  (3),  and  (4)  be  divided  by  A: 


pxcosa+pzxcos  r+pvxcos  P=qx,    ,     ,     .     (7) 

"  py  cos  P+pZy  cos  r+pxv  cos  a  =qy,    .     .     .     (8) 

pz  cos  r  +Pxz  cos  a+pyg  cos  p=qx.     .     .     .     (9) 


If  p  be  the  angle  between  the  axis  of  X  and  the  direc- 
tion of  q,  then  will 

cos  pi  =2^  .......     (I0) 

q 

Eqs.  (8)  and  (9)  give  similar  values  of  the  cosines  of 
the  angles  .between  the  direction  of  q  and  the  axes  of  Y 
and  Z,  thus  fixing  the  direction  of  q. 

Using  the  values  of  qx,  qy,  and  qz  as  given  in  eqs.  (7), 
(8),  and  (9),  the  component  of  q  normal  to  its  plane  of 
action  (ABC  =  J)  will  be: 

qn  =qx  cos  a+qv  cos  p+qg  cos  7.    .     .     .      Cu) 


40  ELASTICITY  IN  AMORPHOUS  SOLID  BODIES.  [Ch.  T. 

Hence  the  obliquity  4>  of  q  can  at  once  be  determined 
by  the  equation 


(12) 


The  triangular  face  XYZ  of  the  tetrahedron  Fig.  i 
may  be  considered  a  part  of  the  exterior  surface  of  a  body 
on  which  acts  the  stress  whose  intensity  is  q.  The  three 
rectangular  coordinates  faces  XOY,  YOZ,  and  ZOX  are 
then  to  be  taken  as  interior  surfaces  of  the  body  on  which 
act  the  internal  stresses  indicated.  The  stress  on  the 
external  face  XYZ  must  be  in  equilibrium  with  the  stresses 
acting  on  the  three  interior  rectangular  coordinate  faces 
of  the  tetrahedron. 

Principal  Stresses  and  Ellipsoid  of  Stress. 

The  preceding  equations  are  general  anfL  relate  to 
stresses  on  any  planes  whatever.  If,  however,  the  stress 
q  is  normal  to  its  plane  of  action  it  is  a  principal  stress. 
In  that  case  the  obliquity  is  zero  and  there  is  no  shear. 
Hence, 


Substituting  these  values  in  the  second  members  of 
eqs.  (7),  (8),  and  (9),  and  then  eliminating  cos  a,  cos  /3, 
and  cos  r  from  the  three  resulting  equations,  the  follow- 
ing equation  of  the  third  degree  will  be  found: 


-  (px  +Pv  +Pz)q2  +  (Pxp 

2xV  -  Pxpypz  ~  2pxvpzxpvz  =0.       .        .        (14) 


Or,  indicating  the  coefficients  of  q  and  the  part  of  this 


Art    ii.]     PRINCIPAL  STRESSES  AND  ELLIPSOID  OF  STRESS.        41 

equation  independent  of  that  quantity  by  A,  B,  and  C, 
respectively  : 

q*-Aq2+Bq-C  =  o  .....      (15)* 

The  three  roots  of  this  cubic  equation  are  the  inten- 
sities of  the  three  principal  stresses,  and  the  equation 
shows  that  at  every  point  three  such  principal  stresses 
exist,  each  normal  to  its  plane  of  action  on  which  there  is 
no  shear. 

If  in.  eq.  (6)  the  coordinate  axes  of  X,  Y,  and  Z  be 
taken  as  the  principal  axes  so  that  the  intensities  px,  pv, 
and  pz  become  the  principal  intensities  qi,  q2,  and  q3, 
then  will  pxy=pyz  =  pxll  =  oy  and  q  will  be  the  intensity 
of  stress  in  any  direction  on  a  plane  whose  normal  makes 
the  angles  a,  0,  and  7  with  the  coordinate  axes,  i.e.,  with 
qi,  q2,  and  qs.  Hence 


cos2  r    .     .     (16) 

Again,  if  qx,  qv,  and  qz  are  the  rectangular  components 
of  q,  Eqs.  (7),  (8),  and  (9),  will,  under  the  same  conditions, 
give  : 

<i  cos  «=*        2cos=          and 


Then,  squaring  and  adding: 


*  Rankine  observed  in  his  Applied  Mechanics  that  if  qit  qz,  and  q3  are 
the  roots  of  a  cubic  equation,  then  : 

(q  -  qi)  (q  —  52)  (q  —  qs)=q3-q2(qi  +52+53)  +5(3132  +325*  +5i5s)  -  qiQzqa  =  o. 

This  equation  shows  that  the  quantities  A,  B,  and  C  remain  the  same 
whatever  may  be  the  directions  of  the  three  rectangular  axes  at  a  given 
point.  Hence,  by  using  A  it  is  seen  that  qi+q*+q>=px+py+pz,  i.e.,  the 
sum  of  the  normal  components  of  the  intensities  of  stress  on  any  three 
rectangular  coordinate  planes  is  constant  and  equal  to  the  sum  of  the 
intensities  of  the  three  principal  stresses. 


42  ELASTICITY  IN  AMORPHOUS  SOLID  BODIES.          [Ch.  I. 

By  dividing  this  equation  through  by  q2  it  may  be 
written  in  terms  of  the  angles  between  q  and  the  coor- 
dinate axes  (eq.  18)  and  the  reciprocal  of  q2. 

Eq.  (17)  is  the  usual  form  of  the  equation  of  an  ellip- 
soid with  the  origin  of  coordinates  at  its  center  in  which 
qi,  q2,  and  q$  are  the  semi  axes  and  qx,  qy  and  qz  are  the 
coordinates  of  any  point  in  the  surface. 

The  intensity  of  stress  q  in  any  direction,  represented 
by  the  semi-diameter  of  the  ellipsoid  in  that  direction, 
is  given  by  eq.  (16),  and  the  angles  between  its  direction 
and  the  coordinate  axes  X,  Y,  and  Z  may,  by  the  aid  of 
eq.  (10),  be  written: 


qx     q  i  cos  a  qv     q-z  cos 

cos  pi  =—  =  —        -;     cosp2=—  =  — 

q       q  q       q 

qz     <?3  cos  r 
3=-=~__ . 


(18) 


The  component  of  q  normal  to  its  plane  of  action  is 
given  by  eq.  (u)  : 


qn  =q\  cos2  a+^2  cos2  fi+qz  cos2  f.        •     •      (T9) 
The  cosine  of  the  obliquity  of  q  is  therefore: 

cos  <£=—  .....     .     .     (20) 

q 

These  elementary  considerations  are  sufficient  for  the 
purpose  of  outlining  to  some  extent  at  least  the  general 
subject  of  stress  in  any  or  all  directions  in  solid  bodies. 
The  results  may  easily  be  developed,  so  as  to  be  applicable 
to  the  solution  of  any  required  problem.  The  equations 
(2),  (3),  and  (4)  are  frequently  applied  to  the  discussion 


Art.  12,]  ELLIPSE  AND  ELLIPSOID  OF  STRAIN  43 

of  the  action  of  external  forces  qx,  qy,  and  qz,  in  connection 
with  the  internal  stresses  px,  pu  and  pz,  etc.,  as  will  be 
indicated  later. 

It  is  obvious  that  if  all  the  internal  stresses  act  parallel 
to  one  plane,  eq.  (14)  and  those  which  follow  it  will  relate 
to  the  ellipse  of  stress,  showing  that  the  latter  is  a  special 
case  of  the  ellipsoid  of  stress. 

Art.  12. — Ellipse  and  Ellipsoid  of  Strain. 

It  has  been  shown  that  the  intensity  of  stress  at  any 
point  in  a  solid  homogeneous  body  may  be  represented 
by  the  semi-diameter  of  an  ellipsoid  in  the  general  case 
or  the  semi-diameter  of  an  ellipse  in  the  special  case  of  all 
stress  being  parallel  to  a  plane.  Inasmuch  as  strains  are 
proportional  to  the  corresponding  stresses  below  the  elastic 
limit,  the  strain  of  a  very  short  but  constant  length  of  a 
solid  element  at  any  point  would  be  represented  by  the 
semi- diameter  of  an  ellipsoid  or  ellipse  having  the  same 
direction  as  the  corresponding  intensity,  which  also  might 
be  represented  by  the  same  semi-diameter  at  a  proper 
scale.  It  follows  from  these  simple  considerations  that 
strains  in  all  directions  may  be  represented  by  ellipsoids 
and  ellipses  as  well  as  stresses.  While  such  ellipsoids  and 
ellipses  possess  analytic  interest  in  connection  with  the 
theory  of  elasticity  in  solid  bodies,  they  are  not  of  sufficient 
importance  in  the  structural  operations  of  engineering  to 
require  even  elementary  analytic  treatment. 

Art.  13. — Orthogonal  Stresses. 

When  stresses  of  tension  or  compression  at  right  angles 
to  each  other  concur  either  in  one  plane  or  on  three  coor- 
dinate planes  making  right  angles  with  each  other,  as  in 


44  ELASTICITY  IN  AMORPHOUS  SOLID  BODIES.  [Ch.  I. 

the  cases  of  the  ellipse  and  ellipsoid  of  stress,  they  are 
said  to  be  orthogonal  stresses.  Such  stresses  produce 
partially  independent  strains  in  the  directions  in  which 
they  act,  but  the  resultant  stress  on  any  one  plane  is  a 
single  stress  obviously  accompanied  by  its  character- 
istic strain.  This  is  true  whether  the  stress  is  wholly 
parallel  to  one  plane  or  if  it  acts  in  all  directions.  The 
fact  that  lateral  and  direct  strains  in  the  same  directions 
may  concur  has  induced  some  engineers  and  writers  to 
attempt  to  provide  rather  arbitrarily  for  the  supposed 
effects  of  orthogonal  stresses  and  strains. 

If  in  the  case  of  stress  wholly  parallel  to  one  plane 
px  and  py  represent  the  intensities  of  the  principal  stresses, 
as  in  Art.  7,  the  unit  strain  parallel  to  the  axis  of  x 
will  be, 


Similarly  the  unit  strain  in  the  direction  of  y  will  be, 


In  the  preceding  eqs.  (i)  and  (2)  the  plus  sign  is  to  be 
used  if  the  intensities  px  and  pv  are  of  opposite  kinds,  but 
the  minus  sign  is  to  be  written  if  the  two  stresses  are  of 
the  same  kind,  i.e.,  both  tension  or  both  compression. 

Two  intensities  of  stress  p'x  and  p'v  are  then  assumed, 
each  of  which  if  acting  separately  would  produce  the 
strains  in  the  two  coordinate  directions,  respectively, 
shown  by  eqs.  (i)  and  (2).  These  two  intensities  must 
have  the  following  values  : 

px±rpv     and     pv±rpx  .....     (3) 


Art.  13.]  ORTHOGONAL  STRESSES.  45 

These  are  called  "  equivalent  "  intensities  of  stress,  and 
it  is  postulated  that  the  working  intensity  of  stress  pre- 
scribed for  any  member  of  a  structure  must  not  exceed 
the  greatest  of  the  two  values  given  by  eq.  (3). 

In  the  special  case  of  two  principal  stresses  being  of 
opposite  kinds  but  of  equal  intensity,  the  greatest  shear 
will  be  of  the  same  intensity  as  the  principal  stresses, 
or  by  the  aid  of  eq.  (3) 

or,  combining  eqs.  (3)  and  (4), 

(s) 


hence, 

P'* 


t+r 

In  the  latter  case  it  is  said  that  the  greatest  shear 
must  not  exceed  pt  in  eq.  (6),  p'x  representing  the  pre- 
scribed working  intensity  in  tension  or  compression  as 
the  case  may  be. 

This  arbitrary  substitution  of  an  intensity  of  stress 
corresponding  to  the  sum  of  two  coordinate  strains,  in  the 
place  of  an  actual  greatest  intensity  of  stress  acting  on  its 
proper  plane,  is  not  supported  by  any  substantial  analytical 
or  experimental  basis.  The  maximum  intensity  of  stress 
at  any  point  in  a  piece  of  material  subjected  to  loading 
may  readily  be  determined  and  the  position  of  the  plane 
on  which  it  acts  may  be  ascertained  by  the  methods  given 
in  the  preceding  articles,  and  it  is  difficult  to  imagine  any 
sufficient  reason  for  not  making  that  actual  maximum 
intensity  of  stress  equal  to  the  prescribed  working  stress 
of  the  same  kind.  The  maximum  intensity  of  stress  at 
any  point  will  of  course  be  accompanied  by  the  maximum 


46  ELASTICITY  IN  AMORPHOUS  SOLID  BODIES.  [Ch.  I. 

unit  strain  and  a  proper  limitation  of  that  strain  will  be 
coincident  with  a  proper  limitation  of  the  stress  pro- 
ducing it.  These  observations  are  equally  true  whether 
the  kind  of  stress  involved  be  tension,  compression,  or 
shearing. 

The  substitution  of  an  artificial  "  equivalent  "  stress, 
therefore,  in  the  place  of  the  actual  maximum  stress  at 
any  point  remains  to  be  justified  and  will  not  be  employed 
in  this  work.  All  the  design  work  involving  the  employ- 
ment of  a  prescribed  working  stress  will  be  based  upon  the 
greatest  actual  intensity  of  stress  in  the  structural  mem- 
ber under  consideration. 

Again,  the  significance  of  lateral  strains  has  been 
expressed  by  stating  that  if  a  straight  bar  of  structural 
steel  with  square  cross-section,  for  illustration,  be  sub- 
jected to  a  tensile  stress  of  intensity  p,  the  lateral  strains 
will  be  negative,  as  they  decrease  the  lateral  dimensions 
of  the  bar,  and  hence  that  if  the  ratio  of  lateral  to  direct 
strains  be  taken  as  one-fourth,  then  those  lateral  strains 
are  each  precisely  the  same  as  would  be  produced  by  an 
intensity  of  compression  equal  to  %p,  acting  at  right  angles 
to  the  bar  and  on  either  pair  of  opposite  sides.  Hence, 
it  has  been  said  that  such  a  bar  is  not  only  subjected  to 
the  axial  tension,  but  also  to  a  "  true  internal  stress  which 
acts  as  a  compression  at  right  angles  to  the  axis  of  the 
bar."  It  is  further  stated  that  such  a  bar  "  suffers  a 
true  internal  compressive  unit  stress  ...  in  all  direc- 
tions at  right  angles  to  its  length  ..." 

It  is  still  further  stated  that  "  The  injury  done  to 
a  body  does  not  depend  upon  the  'actual  stress  or  pres- 
sure, but  upon  the  actual  deformations  produced,  and  the 
true  stresses  are  those  corresponding  to  these  deforma- 
tions." 

It   is    difficult    to   imagine   how    the    "  actual  "    stress 


Art.  13.]  ORTHOGONAL  STRESSES.  47 

existing  at  any  point  in  a  body  fails  to  be  the  "  true  " 
stress.  If  the  "  true  "  stresses  are  different  from  the 
actual  they  must  be  imaginary  or  at  least  not  actual  or 
real. 

It  cannot  be  admitted  that  the  lateral  strains  accom- 
panying the  direct  strains  of  a  bar  subjected  to  axial 
tension  are  produced  by  "  a  true  internal  "  compression, 
for  no  such  corresponding  external  compressive  forces  or 
pressure  at  right  angles  to  the  axis  of  the  bar  exist.  If 
the  lateral  strains  were  due  to  such  compressive  stress, 
the  corresponding  external  compressive  forces  would  per- 
form work  and  would  make  the  total  resilience  of  the  bar 
two-ninths  greater  than  the  resilience  due  to  direct  tensile 
stress  only,  if  the  ratio  r  be  taken  as  one-third. 

This  species  of  confusion  seems  to  arise  at  least  partly 
from  a  failure  to  distinguish  between  molecular  conditions 
below  the  elastic  limit  and  those  above  that  limit. 

If  a  bar  is  subjected  to  axial  tension  producing  corre- 
sponding axial  and  lateral  strains,  in  consequence  of  which 
the  lateral  dimensions  of  the  bar  decrease,  it  by  no  means 
follows  that  actual  compression  has  produced  that  decrease. 
In  fact,  since  the  molecules  have  been  separated  to  a  slight 
.degree  axially,  the  transverse  movement  of  the  molecules 
may  easily  be  conceived  to  take  place  without  any  com- 
pression whatever,  and  the  fact  that  the  density  of  the 
material  is  decreased  by  tensile  stress  makes  that  view 
reasonable,  and  perhaps  conclusively  confirms  it.  It  should 
be  remembered  that  all  these  analytic  investigations  re- 
late only  to  stresses  and  strains  existing  below  the  elastic 
limit. 

While  it  is  true  that  experimental  investigation  is 
still  lacking  to  give  complete  information  regarding  the 
effects  of  orthogonal  stresses  and  strains  below  the  elastic 
limit  (as  well  as  above  it)  there  is  lacking  material  evi- 


48  ELASTICITY  IN  AMORPHOUS  SOLID    BODIES.          [Ch.  I. 

dence  showing  the  existence  of  any  such  stress  conditions 
consequent  upon  the  existence  of  lateral  strains  as  those 
to  which  allusions  are  made  above,  and  they  will  not  be 
recognized  in  the  analytic  work  which  is  to  follow. 

In  discussing  the  stresses  in  the  walls  of  thick  cylin- 
ders in  Appendix  II,  the  bearing  of  these  considerations 
on  the  formula  of  Clavarino  will  be  fully  set  forth. 


PROBLEMS  FOR  CHAPTER  I. 

Problem  i. — A  wrought  iron  bar  4//x|//  in  section  is 
subjected  to  a  tensile  force  of  28,000  pounds.  The  stretch 
for  a  gaged  length  of  20  feet  was  0.12  inch.  Find  the 
intensity  of  tensile  stress  in  the  material,  the  modulus  of 
elasticity  E,  and  the  rate  of  strain,  i.e.,  the  strain  per 
linear  inch. 

Ans.  Intensity  of  stress  =  14,000  Ibs.  per  square  inch. 
£  =  28,000,000  pounds  per  square  inch. 
Rate  of  strain  =  0.0005  inch  per  inch. 

Problem  2. — A  steel  eye-bar  8//X2//  in  section  carries 
a  total  load  of  128,000  pounds,  under  which  there  is  a 
stretch  of  0.016"  in  a  gaged  length  of  5  ft.  Find  the  in- 
tensity of  stress,  rate  of  strain,  and  modulus  of  elas- 
ticity E. 

Problem  3. — Steel  has  a  modulus  of  elasticity  of 
30,000,000  pounds  per  square  inch,  and  a  coefficient  of 
expansion  of  0.0000065  per  degree  F.  If  a  steel  bar  2"X4" 
in  cross-section  has  a  length  of  30'  o"  at  a  temperature 
of  40°  F.,  find  the  length  of  the  bar  at  10°  F.  and  at  110° 
F.  Suppose  the  ends  of  this  bar  had  been  fastened  rigidly 
at  the  temperature  of  40°  F.  Find  the  intensity  of  ten- 
sile stress  at  10°  F.  and  intensity  of  compressive  stress  at 


Art.  13.]  PROBLEMS  FOR   CHAPTER   I.  480 

110°  F.,  supposing  the  bar  to  be  firmly  held  against  lateral 
deflection. 

Partial  Ans.  Length  of  bar  at  10°  F.  =29'. 99415. 

Intensity  of  tensile  stress  in  bar  at  a 
temperature  of  10°  F.=585o  pounds 
per  square  inch. 

Problem  4. — A  concrete  pillar  24^X2^'  in  section  and 
8  ft.  high  carries  a  total  (compressive)  load  of  115,200 
pounds.  If  the  modulus  of  elasticity  for  the  concrete 
is  2,500,000  pounds  per  square  inch,  what  will  be  the 
rate  of  compressive  strain  and  the  shortening,  first,  for 
the  total  height  8  ft.  of  pillar,  and,  second,  for  12",  under 
the  preceding  load? 

Problem  5. — In  Problems  i  and  2,  if  Poisson's  ratio 
r  (i.e.,  the  ratio  of  lateral  to  direct  strain)  is  0.3,  find  the 
new  cross-dimension  of  the  bars  and  also  the  change  in 
volume  for  a  portion  of  each  bar  i  foot  long. 

Ans.  for  Problem  i. 

^  =  3".  99916;  b  =0^.499895; 

change  in  volume  =  0.00908  cubic  inch  decrease. 

Problem  6. — In  Problem  3,  the  cross-dimensions  of 
the  bar,  under  the  compressive  stress,  become  2". 000114 
and  4". 0002 2 8.  Find  the  ratio  r  between  direct  and 
lateral  unit  strains,  and  also  the  increase  of  volume  of  3 
ft.  length  of  the  bar. 

Problem  7. — In  Problems  5  and  6  find  the  modulus 
of  elasticity,  G,  for  shearing  in  terms  of^  the  direct  modulus 
of  elasticity  E. 

Problem  8. — In  Problem  2  find  the  total  normal  and 
tangential  stresses  and  their  intensities  on  plane  sections 
making  angles  of  18°,  35°,  and  53°  with  the  axis  of  the 
piece. 

Problem  9. — In  Problem  3  find  the  total  normal  and 
tangential  stresses  and  their  intensities  on  plane  sections 


486  ELASTICITY  IN  AMORPHOUS  SOLID   BODIES.  [Ch.  I. 

making  angles  of  31°,  45°,  and  72°  with  the  axis  of  the 
piece. 

Problem  10. — A  round  steel  bar  3  inches  in  diameter 
is  subjected  to  a  tensile  stress  of  212,100  pounds.  If  the 
diameter  of  the  bar  decreases  0.00105  inch,  find  the 
ratio  r  between  the  direct  and  lateral  strains,  and  also 
the  increase  of  volume  in  a  4-ft.  length  of  bar.  Assume 
modulus  of  elasticity  E  as  30,000,000  pounds  per  square 
inch. 

Problem  n. — Given  three  planes,  AO,  OB,  and  BA, 
Art.  8,  so  placed  that  AOB  =90°  and  ABO  =  a  =  6o°. 

The  tensile  stress  on  OB  is  £3=3500  pounds  per  square 
inch  and  the  tensile  stress  on  OA,  ^  =  5600  pounds  per 
square  inch.  The  shearing  stresses  on  OA  and  OB  are 
equal,  i.e.,  pxv  =  pvx  =  1750  pounds  pen  square  inch. 

Find  the  normal  and  tangential  components  of  the 
resultant  intensity  £,  when  p  makes  an  angle  0  =  io°, 
below  the  normal  to  the  plane  AB.  Also  find  the  inten- 
sity of  the  principal  stress  on  the  plane  AB, 

Problem  12. — In  Fig.  i,  Art.  8,  let  the  intensity  of  the 
normal  tensile  stress  on  the  plane  OB  be  8000  pounds  per 
square  inch,  while  the  intensity  of  normal  compressive 
stress  on  the  plane  OA  is  12,000  pounds  per  square  inch, 
and  let  the  intensities  of  shearing  stresses  on  the  same 
planes  OB  and  OA  be  3500  and  6500  pounds  per  square 
inch  respectively.  Find  the  principal  stresses  and  the 
principal  planes  on  which  they  act.  Then,  by  means  of 
the  formula  of  Art.  9,  find  the  greatest  intensity  of  shear- 
ing stress  on  any  plane  at  0,  and  the  position  of  that  plane. 
Finally,  determine  the  intensity  of  the  stress  of  greatest 
obliquity  at  the  point  0,  and  the  plane  on  which  it  acts, 
together  with  the  intensity  of  shearing  stress  on  that 
plane. 


CHAPTER  II. 
FLEXURE. 

Art.  14. — The  Common  Theory  of  Flexure. 

A  STRAIGHT  piece  or  bar  of  material  is  subjected  to 
flexure  or  bending  when  it  is  acted  upon  by  loads  or  forces 


at  right  angles  to  its  axis,  the  loads  and  supporting  forces 
taken  as  a   whole   constituting  a   system   in   equilibrium. 

49 


50  FLEXURE.  [Ch.  II. 

The  beam   shown  in   Fig.    i   may  be  taken  to  illustrate 
the  general  condition  of  flexure  or  bending. 

Each  end  of  the  beam  is  supported  as  shown  at  R  and 
Rf,  the  reactions  at  those  points  constituting  the  support- 
ing forces,  while  the  weights  W1  and  VV2,  etc.,  constitute 
the  loading.  The  reactions  are  in  reality  just  as  much 
loads  on  the  beam  as  the  weights  carried  by  it,  but  it  is 
convenient  always  to  make  the  distinction  between  loads 
and  reactions  or  supporting  forces. 


6 


FIG.  2. 

An  overhanging  beam  is  shown  in  Fig.  2  carrying  the 
weights  Wl  and  W2,  etc.,  one  end  being  firmly  fixed  in  a 
wall  or  other  similar  supporting  mass.  In  this  case  the 
supporting  effect  of  the  material  in  which  one  end  of  the 
beam  is  embedded  is  equivalent  to  the  couple  whose 
moment  is  Fe.  Obviously  there  may  be  many  other 
different  cases  of  bending,  according  to  the  manner  of 
supporting  and  loading  the  bent  piece  or  beam. 

In  all  these  analyses  and  in  all  that  follow,  except 
when  otherwise  specially  noted,  the  beams  are  supposed 
to  be  horizontal  with  the  loads  and  reactions  vertical,  all 
external  forces  thus  acting  at  right  angles  to  the  axis  of 
the  beam,  and  they  are  further  supposed  to  lie  all  in  a 
vertical  plane  passing  through  the  same  axis.  When  the 
loading  acts  as  shown  in  Fig.  i ,  it  is  evident  that  the  beam 


Art.  14.]  THE  COMMON    THEORY  OF  FLEXURE.  5l 

will  be  bent  so  as  to  become  convex  downward  and  con- 
cave upward,  thus  causing  the  upper  portion  of  the  beam 
to  be  in  compression  while  the  lower  portion  is  in  tension. 
Hence  if  any  normal  section  of  the  beam  as  BD  be  con- 
sidered, in  passing  from  B  where  there  is  compression 
to  D  where  there  is  an  opposite  stress  of  tension  it  is  clear 
that  at  some  point,  as  m,  there  will  be  a  zero  stress,  or, 
in  other  words,  no  stress  at  all.  The  horizontal  line  pass- 
ing through  that  point  m  of  no  stress,  and  normal  to  the 
vertical  plane  through  the  axis  of  the  beam,  is  called  the 
neutral  axis  of  the  section  and  its  locus  HX  throughout 
the  entire  beam  is  called  the  neutral  surface.  On  one  side 
of  the  neutral  axis  in  any  normal  section  there  will  be 
direct  stresses  of  compression  and  on  the  other  direct 
stresses  of  tension.  There  are  two  fundamental  assump- 
tions in  the  common  theory  of  flexure: 

First,  that  all  plane  normal  sections  of  the  beam  remain 
plane  after  flexure  or  bending. 

Second,  that  the  intensity  (amount  uniformly  dis- 
tributed on  a  square  unit)  of  either  the  tensile  or 
compressive  stress  in  any  normal  section  acting 
parallel  to  the  axis  of  the  beam  varies  directly  as 
the  distance  from  the  neutral  axis  of  the  section. 

In  Fig.  i  the  shaded  triangles \above  and  below  m, 
having  the  common  vertex  at  that  point,  represent  the 
stresses  of  tension  and  compression  induced  in  the  normal 
section  BD  by  the  bending. 

The  loads  and  supporting  forces  act  normally  to  the 
axis  of  the  beam  upon  either  portion  of  it,  as  HBD,  while 
the  internal  stresses  of  tension  and  compression  in  the 
section  BD  act  parallel  to  that  axis.  If  the  equilibrium 
of  the  same  portion  HBD  be  considered,  it  will  be  seen 
that  the  only  horizontal  forces  acting  upon  it  are  the  in- 


52  FLEXURE.  [Ch.  II. 

te-rnal  stresses  of  tension  and  compression  shown  by  the 
two  shaded  triangles.  Hence  in  ord°r  that  there  may 
be  equilibrium  the  sum  of  those  stresses  of  tension  and 
compression  must  be  equal  to  zero.  This  latter  condition 
will  determine  in  a  simple  manner  the  position  of  the 
neutral  axis.  If  a  is  the  intensity  of  either  the  tensile  or 
compressive  stress  at  the  distance  unity  from  the  neutral 
axis,  then  by  the  second  of  the  preceding  fundamental 
assumptions  the  intensity  N,  at  any  other  distance  z  from 
the  same  axis  or  line  of  no  stress,  will  be  N  =  az.  Again, 
if  A  is  the  area  of  the  normal  section  of  the  beam,  dA  will 
be  the  area  of  an  indefinitely  small  portion  of  that  section, 
so  that  the  amount  of  internal  stress  acting  on  it  will  be 
az.dA.  If  this  differential  amount  of  stress  be  integrated 
for  the  entire  section,  the  preceding  condition  of  equilibrium 
for  either  portion  of  the  beam  requires  that  the  sum  repre- 
sented by  that  total  integration  shall  be  equal  to  zero; 
or  if  dl  and  d  represent  the  distances  of  the  most  remote 
fibres  on  either  side  of  the  neutral  axis,  the  following 
equations  may  be  written: 


/di  /V, 

azdA  =a  I     zdA  =o, 
d  J-d 


or 

rdl 

zdA  =o (i) 

Eq.  (i)  shows  that  the  static  moment  of  the  entire 
section  about  the  neutral  axis  is  equal  to  zero,  and  there- 
fore that  the  neutral  axis  passes  through  the  centre  of 
gravity  or  the  centroid  of  the  normal  section. 

It  is  next  necessary  to  determine  the  expression  for 
the  bending  moment  of  the  internal  stresses  of  any  sec- 
tion, such  as  JF  of  Fig.  i ,  which  is  induced  by  and  must 


Art.  14.]  THE   COMMON   THEORY  OF  FLEXURE.  53 

be  equal  to  the  moment  of  the  external  forces  acting  upon 
cither  one  of  the  two  portions  into  which  the  beam  is 
divided  by  that  section. 

In  Fig.  i,  let  mn  represent  a  differential  length,  dl 
of  the  neutral  surface,  and  let  p  represent  the  radius  of 
curvature  of  dl  after  flexure,  also  as  shown  in  Fig.  i,  C 
being  the  centre  of  curvature.  If  u  is  the  direct  or  longi- 
tudinal strain  of  a  unit  length  of  fibre  at  the  distance  unity 
from  the  neutral  axis,  when  stressed  with  the  intensity  a, 
the  strain  in  dl  under  that  intensity  will  be  udl.  BD  is 
drawn  parallel  to  JF,  and  represents  the  position  of  BD 
before  flexure.  The  triangle  D'mD  k,  therefore,  similar 
to  Cmn.  Consequently  there  may  be  written 

.   udl     dl  i 


Or  the  rate  of  strain,  i.e.,  the  strain  of  a  unit'  length  of 
fibre  at  distance  unity  from  the  neutral  axis,  is  equal  to 
the  reciprocal  of  the  radius  of  curvature. 

By  the  fundamental  law  or  assumption  of  the  common 
theory  of  flexure  already  given 

z 
Rate  of  strain  at  distance  z  =  ~. 

Then,  by  the  fundamental  law  between  stress  and 
strain,  the  intensity  N  of  the  direct  stress  at  any  distance 
z  is 


(3) 


If  b  is  the  variable  breadth  of  section,  the  differential 
of  the  total  stress  is 

Nbdz=-.(bdz).z  ......     (4) 


54  FLEXURE.  [Ch.  II. 

The  differential  moment  of  the  internal  stresses  about 
the  neutral  axis  will  be 

=  -.(bdz).z2;  ....     (5) 

FT 

fofe).*2=— ;     ....     (6) 
P 

in  which  I  is  the  moment  of  inertia  of  the  section  of  the 
beam  about  the  neutral  axis. 

If  x  is  the  horizontal  coordinate  of  the  neutral  sur- 
face, and  w  the  deflection  or  sag  of  the  beam  at  any  point, 
as  indicated  in  Fig.  i ,  when  the  curvature  is  small 


and 


Eq.  (7)  is  the  fundamental  equation  by  which  the  de- 
flection of  a  bent  beam  is  found,  whatever  may  be  the 
character  or  amount  of  the  loading.  As  indicated,  it  is 
strictly  true  only  when  the  deflections  are  small ;  in  other 
words,  when  they  are  produced  by  strains  within  the  elastic 
limit  of  such  beams  as  are  ordinarily  used  in  engineering 
practice.  That  equation  is  easily  integrated  in  all  ordi- 
nary cases,  if  the  value  of  the  external  bending  moment  M 
is  expressed  in  terms  of  x,  as  will  be  abundantly  illustrated 
in  succeeding  articles. 

Another  equation  of  great  practical  value  remains  to 
be  established.  Let  it  first  be  observed  that  the  intensity 
of  stress  a,  at  the  distance  of  unity  from  the  neutral  sur- 


Art.  14.]  THE  COMMON   THEORY  OF  FLEXURE.  55 

face  of  a  bent  beam  is  a  =Eu,  by  Hooke  's  law,  and  further 
by  eq.  (2) 

a=Eu=  —  .......      (g) 

£ 

If  the  value  of  —  from  eq.  (8)  be  substituted  in  eq.     (6) 

there  will  result 

M=aL      .......      (9) 

If  the  greatest  intensity  of  stress  in  a  normal  section 
of  a  bent  beam  at  the  distance  dt  from  the  neutral  axis  be 

k 
represented  by  k,  then  a  =  r,  and  eq.  (9)  will  take  the  form 


do) 


Eq.  (10)  is  one  of  the  most  important  equations  in  the 
whole  subject  of  the  resistance  of  materials  in  consequence 
of  its  frequent  use  in  the  practical  operation  of  designing 
beams  or  girders.  Its  employment  is  rendered  exceed 
ingly  simple  and  convenient  by  tables  in  which  may  be 
found  computed  the  moments  of  inertia  7  for  all  the  rolled 

sections,  as  well  as  values  of  the  quantity  ^-,  called  the 

"  section  modulus."  These  tables  are  found  in  the  various 
"  Hand-books"  published  by  steel-producing  companies, 
and  they  obviate  essentially  all  numerical  computations 
for  the  determination  of  either  moment  of  inertia  or  section 
modulus.  Other  tables  may  also  be  found  which  give  the 
moments  of  inertia  of  a  great  variety  of  built  sections, 
i.e.,  composite  sections  formed  of  various  commercial 
rolled  shapes  such  as  plates,  angles,  channels,  and  I  beams. 
In  all  the  preceding  expressions  where  the  quantity 


56  FLEXURE.  [Ch.  II. 

M  appears  it  is  to  be  taken  to  represent  the  bending  mo- 
ment of  the  external  forces,  including  the  reactions,  applied 
to  a  beam,  the  moment  being  taken  about  the  neutral 
axis  of  the  section  under  consideration.  This  external 
moment  must  necessarily  be  equal  to.  the  moment  of  the 
internal  stresses  represented  by  the  last  members  of  the 
preceding  moment  equations  involving  the  greatest  in- 
tensity of  stress  k  of  the  section  and  the  moment  of  inertia 
/  of  the  latter. 

There  are  one  or  two  approximate  features  involved 
in  the  preceding  analysis,  the  character  of  which  is  not 
discoverable  when  the  fundamental  laws  of  the  theory  of 
flexure  are  assumed  rather  than  demonstrated,  but  which 
appear  plainly  evident  in  the  true  demonstration  of  the 
theory  of  flexure  in  App.  I.  It  is  obvious  that  the  com- 
pression produced  at  the  exterior  surface  of  a  bent  beam 
at  the  points  of  loading  is  neglected  or  ignored  in  the  pre- 
ceding demonstrations;  but  this  does  not  sensibly  affect 
the  accuracy  of  the  formulas  which  have  been  reached. 
There  is,  however,  one  result  of  the  assumptions  made 
which  materially  affects  the  accuracy  of  the  formulas  of  the 
common  theory  of  flexure  for  comparatively  short  beams. 
If  the  accurate  analysis  be  followed  it  will  be  found  that 
the  formulae  of  that  theory  involve  in  reality  the  further 
assumption  that  the  depth  of  the  beam,  i.e.,  in  the  direc- 
tion of  the  loading,  is  small  in  comparison  with  the  length 
of  span.  The  limit  of  ratio  of  length  of  span  to  depth 
above  which  the  formulae  may  be  applied  with  strict  accu- 
racy cannot  be  definitely  assigned,  but  there  are  many 
beams,  especially  of  timber,  employed  in  engineering 
practice  which  are  much  too  short  in  comparison  with 
their  depths  to  permit  an  accurate  application  of  the  for- 
mulae of  the  common  theory  of  flexure.  This  observation 
bears  with  special  emphasis  on  computations  for  pins  in 


Art,  15.]  THE  DISTRIBUTION  OF  SHEARING  STRESS.  57 

pin-connected  bridges  which  are  treated  as  short  beams. 
As  a  matter  of  fact,  the  common  theory  of  flexure  cannot 
be  applied  to  such  short  thick  beams  with  any  degree  of 
accuracy  whatever.  It  is,  however,  entirely  permissible 
to  use  these  formulae  as  general  expressions,  even  under 
such  loosely  approximate  conditions,  into  which  empirical 
quantities  established  under  the  actual  conditions  of  use 
are  introduced,  but  they  are  not  to  be  used  in  any  other 
way.  By  such  a  procedure  the  formulae  of  the  common 
theory  of  flexure  have  become  of  inestimable  value  to  the 
civil  engineer,  but  it  is  imperative  to  realize  under  what 
conditions  they  may  be  employed  with  strict  accuracy 
and  under  what  conditions  the  introduction  of  quantities 
established  by  practical  tests  is  required. 

Art.    15. — The  Distribution  of  Shearing  Stress  in  the  Normal 
Section  of  a  Bent  Beam. 

The  longitudinal  fibres  of  a  beam  under  loading  take 
their  stresses  of  tension  and  compression  from  the  shearing 
stresses  which  are  induced  on  vertical  and  horizontal 
planes  in  the  interior  of  the  beam.  In  order  to  realize 
what  takes  place  in  the  interior  of  a  beam  let  it  be  sup- 
posed to  be  divided  into  an  indefinitely  large  number  of 
small  rectangular  portions  like  those  shown  in  the  up- 
per part  of  Fig.  i,  and  on  a  somewhat  larger  scale  in  the 
lower  part.  The  vertical  loading  and  reactions  induce 
transverse  shears,  i.e.,  shearing  stresses  on  vertical  trans- 
verse planes,  which,  as  known  from  the  general  theory  of 
stresses  in  solid  bodies,  induce  shears  of  equal  intensity  on 
horizontal  planes.  The  result  is  that  which  is  shown  in 
the  lower  portion  of  Fig.  i.  On  the  faces  of  the  indefi- 
nitely small  rectangular  portions  of  the  beam  there  are 
induced  shears  in  pairs  having  the  same  intensity  and  act- 


FLEXURE. 


[Ch.  II. 


ing  either  toward  or  from  a  given  edge.  Each  horizontal 
layer  of  the  beam  is,  therefore,  made  to  slide  a  little  over 
the  adjoining  layers  above  and  below  it,  as  shown  at  A  and 
A'  in  the  lower  part  of  Fig.  i. 


FIG.  i. 

Carefully  remembering  these  general  conditions,  let  the 
bending  moment  in  the  section  ad  of  the  beam  in  Fig.  2 
be  represented  by  M  and  let  the  total  transverse  shear  at 


FIG.  2. 

the  same  section  be  represented  by  5.     Then  if  x  measured 

M 
horizontally  from  the  section  ad  be  so  taken  that  x=-^t 


Art.  15.]          THE  DISTRIBUTION  OF  SHEARING  STRESS.  59 

and  if  the  intensity  of  the  direct  stress  of  tension  or  com- 
pression at  the  distance  z  from  the  neutral  axis  be  repre- 
sented by  kt  there  may  at  once  be  written 

TI/T     o       kl  L     $z  i  \ 

M=Sx=  —  ;     ..  k=-j-x  .....     (i) 
z  J. 

k  is  thus  seen  to  be  a  function  of  both  z  and  x.  If  z  be 
unchanged  while  x  varies,  the  small  variation  of  k  for  an 
indefinitely  small  variation  of  x  will  be 

dk        Sz 

•     .....     (2) 


If  5  is  the  intensity  of  the  transverse  shear  at  the  dis- 
tance z  from  the  neutral  axis,  the  variation  of  that  intensity 
for  the  indefinitely  short  distance  dz  (x  remaining  unchanged) 

will  be  ~rdz,  and  if  the  breadth  or  width  of  the  beam  is  b, 

dz 

the  variation  of  longitudinal  shear  on  the  small  horizontal 
area  bdx  for  the  small  distance  dz  will  be 


(3) 


The  small  shear  given  by  expression  (3)  is  equal  to  the 
variation  of  k  found  by  multiplying  the  members  of  eq.  (2  ) 
by  bdzt  hence 

ds  Sz 

.....     (4) 


ds     Sz  5 

••  &-y   or  ds=7zd*  .....    (s) 

It  is  obvious  that  the  intensity  of  the  shear  at  the  ex- 
terior surface  of  the  beam  is  zero;  in  other  words,  s  =  o, 
when  z  =  d  the  distance  of  the  extreme  fibre  of  the  section 


60  FLEXURE.  [Ch.  II. 

from  the  neutral  axis.  Hence  eq.  (5)  must  be  integrated 
between  the  limits  of  z  and  d,  and  that  integration  wiD 
give 


*  The  intensity  of  shear  s  is  sometimes  found  with  a  partial  regard  only 
to  the  laws  of  the  Common  Theory  of  Flexure.  In  Fig.  3  the  piece  abed  of 
a  beam  subjected  to  flexure  whose  neutral  surface  is  NN  is  held  in  equilib- 
rium by  the  direct  stresses  on  the  faces  be  and  ad  in  combination  with  the 
longitudinal  shear  on  the  face  dc.  If  ab  is  equal  to  dx  and  if  y  be  the  normal 
distance  of  any  fibre  from  NN,  obviously  the  difference  between  the  direct 

stresses  on  the  two  sides  be  and  ad  will  be  I       dk.bdy  in  which  b  is  the  vari- 

Jy 
able  width   of  the  section.     By  the  common  theory  of  flexure,  however, 

dM  dM  A/i  . 

dk  —  —f-y-     Hence  the  above  expression  becomes  —=-        ybdy.      If    s  is    the 

intensity  of  shear  on  the  face  dc  the  following  equation  at  once  results: 

a  b 


dM  C 

=  ~Y~ 

*  Jy 


FIG.  3. 

v  i 

ybdy,     ...........     (a) 


This  equation  differs  from  eq.  (6)  in  that  b,  considered  as  a  variable, 
appears  in  the  second  member.  If  the  section  is  rectangular,  b  is  constant 
and  eq.  (6)  at  once  results.  In  fact  if  yi  and  y  be  taken  as  consecutive  in 
eq.  (a),  which  is  the  differential  method  of  establishing  s,  that  equation  will  be- 
come 

dsbdx  =  ~Y~ybdy. 

The  quantity  b  now  disappears  from  the  equation  whether  the  width  of  the 
section  be  considered  constant  or  variable.  Then  dividing  both  sides  of  the 


Art.  15.]         THE  DISTRIBUTION  OF  SHEARING  STRESS.  61* 

The  intensity  s  has  its  maximum  value  where  z=o,  i.e., 
at  the  neutral  axis;   hence 

Sd2 
(max.)  5=—^ (7) 


Sbd3 
If  the  section  is  rectangular  /  = - — 


and 


In  other  words,  the  maximum  intensity  of  shear  found  at 
the  neutral  axis  is  -,  the  average  shear  of  the  entire  section. 

It  is  to  be  remembered  that  this  intensity  of  shear  s, 
at  all  points  in  the  entire  beam,  acts  on  both  the  vertical 
and  horizontal  planes,  i.e.,  this  shear  acts  on  longitudinal 
or  horizontal  planes  parallel  to  the  neutral  surface  as  well 
as  upon  the  vertical  section  of  the  beam. 

Eq.  (6)  is  the  equation  of  a  parabola  with  its  vertex 
in  the  neutral  surface.  Hence  if  Of  be  laid  off,  as  shown 
in  Fig.  2,  at  any  convenient  scale  to  represent  the  maxi- 
mum value  of  s,  as  given  in  eq.  (7),  and  if  from  /  as  ver- 
tices the  two  branches  of  parabolic  curves  fa  and  fd  be 
described  as  shown,  any  horizontal  abscissa  of  the  curves 
drawn  from  the  line  ad  will  represent  the  intensity  of  shear 
at  that  point.  The  origin  of  coordinates  for  eq.  (6)  is  at 
0  in  Fig.  2. 

equation  by  dx  and  integrating,  eq.  (6)  of  the  text  will  be  established.  This 
means  that  all  fibres  equidistant  .from  the  neutral  axis  being  stressed 
uniformly  and  hence  without  longitudinal  shear  along  their  vertical  sides, 
the  beam  may  be  considered,  so  far  as  this  analysis  is  concerned,  as  com- 
posed of  vertical  rectangular  strips  of  width_6,  which  may  be  of  finite  value 
or  indefinitely  small. 


62 


FLEXURE. 


[Ch.  II. 


Distribution  of  Shear  in  Circular  and  Other  Sections. 

A  number  of  special  approximate  investigations  have 
been  made  to  determine  the  distribution  of  shear  in  the 
circular  cross-section  of  a  bent  beam,  involving  more  or 
less  complicated  consideration  of  stresses.  While  these 
investigations  recognize  the  straight  line  variation  of  the 
intensities  of  normal  stresses  in  the  section  under  con- 
sideration, they  are  based  on  other  conditions  which  are 


FIG.  4. 

not  closely  consistent  with  the  fundamental  assumptions 
of  the  Common  Theory  of  Flexure. 

If  the  intensity  of  normal  stress  is  the  same  at  a  uni- 
form distance  from  the  neutral  axis  of  the  section,  adjacent 
fibres  equidistant  from  that  axis  will  stretch  the  same 
amount,  eliminating  all  shearing  stresses  between  such 
fibres.  If  therefore  a  circular  section  whose  area  is  A 
be  divided  into  vertical  strips  each  with  the  width  dy 
as  shown  in  Fig.  4,  and  if  the  notation  shown  in  that 
figure  be  observed,  eq.  (6)  may  be  adapted  to  the  circular 
section  by  placing  in  the  second  member  of  that  equation, 

5     .      for  5  and dy  for  I,  resulting  as  follows: 


-riU-^ ,    (9) 


Art.  15.]  DISTRIBUTION  OF  SHEAR.  63 

This  equation  gives  the  value  of  the  intensity  of  shear 
in  all  parts  of  the  circular  section.  If  z=d,  i.e.,  at  all  points 
of  the  surface,  the  intensity  5  is  zero.  The  maximum 

intensity  is  found  by  making  z  =  o,  giving  s=— 3,  i-e->  the 

maximum  intensity  of  shear  is  |  the  mean,  as  was  to  be 
expected.  The  same  result  will  necessarily  follow  the 
same  mode  of  treatment  of  any  form  of  section  whatever^ 
as  each  such  section  is  assumed  to  be  made  up  of  vertical 
rectangular  strips  between  which  no  shear  exists.  The 
difference  between  this  simple  approximate  method  based 
upon  results  for  a  rectangular  section  and  one  of  the 
special  analyses  for  a  circular  section  is  shown  by  the 
maximum  intensity  of  shearing  stress  at  the  neutral  sur- 
face being  found  equal  to  f  (instead  of  f )  of  the  mean  by 
one  of  those  special  methods.  If,  however,  the  ordinary 
assumptions  of  the  Common  Theory  of  Flexure  are  to  be 
made  at  all  the  advantage  or  increased  accuracy  of  such 
special  or  more  complicated  analyses  is  not  obvious. 

With  such  material  as  timber,  in  the  case  of  beams, 
the  longitudinal  shear  represented  by  s  in  either  eq.  .(7) 
or  eq.  (8)  may  be  the  governing  quantity  in  design.  The 
capacity  of  timber  to  resist  shear  along  its  fibres  is  com- 
paratively so  small  that  where  the  spans  are  relatively 
short  failure  will  take  place  by  shearing  along  the  neutral 
surface  before  the  extreme  fibres  yield  either  in  tension 
or  compression.  In  the  design  of  timber  beams,  there- 
fore, and  in  other  similar  cases,  it  is  necessary  to  test  by 
computation,  the  maximum  value  of  s  as  well  as  to  deter- 
mine the  greatest  intensity  of  tensile  or  compressive  stress 
in  the  extreme  fibres,  as  will  be  completely  shown  in  a 
later  article. 


64 


FLEXURE. 


[Ch.  II. 


Art.  16. — External  Bending  Moments  and  Shears  in  General. 

Beams  subjected  to  pure  bending  only  will  be  treated 
here. 

A  beam  is  said  to  be  non-continuous  if  its  extremities 
simply  rest  at  each  end  of  the  span  and  suffer  no  constraint 
whatever. 

A  beam  is  said  to  be  continuous  if  its  length  is  equal 
to  two  or  more  spans,  or  if  its  ends,  in  case  of  one  span  (or 
more)  suffer  constraint. 

A  cantilever  is  a  beam  which  overhangs  its  span,  one 
end  of  which  is  in  no  manner  supported.  Each  of  the 
overhanging  portions  of  an  open  swing  bridge  is  a  canti- 
lever truss. 


K — *IT -*! 

2 


-  d  C)  o 


'----. 

FIG.  i. 


-------  f 

1 


Fig.  i  represents  a  beam  simply  supported  at  each  end, 
carrying  the  loads  Wlt  W^  Ws,  etc.  Let  bending  moments 
be  taken  for  any  section,  as  JF,  at  the  distance  x'  from 
the  right-hand  abutment,  at  which  location  the  reaction 
R'  acts.  The  load  W^  is  at  the  distance  xl  from  the  sec- 
tion,  W2  a,t  the  distance  x2,  and  W3  at  the  distance  xs  from 
the  same  section,  the  last  distance  not  being  shown  in 
the  figure.  The  bending  moment  desired  will  be  the 
following  : 

.     .     .     (i) 


Art.  16.]      EXTERNAL  BENDING  MOMENTS  AND  SHEARS.  65 

This  equation  is  typical  of  all  external  bending  moments 
for  a  beam  simply  supported  at  each  end,  whatever  may 
be  the  system  of  loading  or  its  amount,  or  whatever  may 
be  the  location  of  the  section.  This  equation  is  frequently 
written  in  the  following  form  : 


(2) 


The  summation  sign  indicates  that  the  sum  is  to  be 
taken  of  the  products  formed  by  multiplying  each  external 
force,  whether  loading  or  reaction,  by  its  lever-arm  or 
normal  distance  from  the  section  in  question.  It  is  a 
common  and  convenient  mode  of  expressing  the  general 
value  of  the  bending  moment  in  any  case  whatever. 

In  eq.  (i)  the  differentials  of  x'  ,  xlt  x2,  and  x3  are  all 
equal,  so  that  if  that  equation  be  differentiated,  the  first 
derivative  of  M  will  have  the  following  form  :  « 


S.        .     .     (3) 

It  will  be  at  once  evident  that  5  in  eq.  (3)  is  the  total 
transverse  shear  in  the  section  for  which  the  bending 
moment  M  is  written,  since  the  algebraic  sum  of  R'  and  the 
loads  between  the  end  of  the  beam  and  the  section  con- 
stitutes that  shear.  Indeed,  the  usual  manner  of  deter- 
mining the  total  transverse  shear  is  the  simple  operation 
of  summing  up  all  the  external  forces  acting  on  one  of  the 
portions  of  the  beam  formed  by  the  section  in  question; 
the  external  forces,  such  as  the  reaction,  acting  in  one 
direction  being  given  one  sign,  and  those,  like  the  loading, 
acting  in  the  other  direction  being  given  the  opposite  sign. 
The  shear,  therefore,  becomes  the  numerical  difference 
of  the  two  sets  of  forces  having  opposite  directions. 

Eq.  (3)  thus  establishes  the  following  important  prin- 
ciple: The  total  transverse  shear  at  any  section  is  equal 


66  FLEXURE.  [Ch.  II. 

to  the  first  differential  coefficient  of  the  bending  moment  con- 
sidered a  junction  of  x. 

In  Fig.  i  the  force  5  is  supposed  to  be  the  resultant  of 
the  three  loads  Wv  W2,  and  WB,  and  the  reaction  R',  i.e., 
the  force  5  is  supposed  to  represent  that  resultant  both 
in  line  of  action  and  magnitude.  The  bending  moment  M 
is,  therefore,  equal  to  Se,  e  being  the  normal  distance  of 
the  line  of  action  of  5  from  the  section,  so  that  the  actual 
bending  moment  upon  any  section  of  a  bent  beam  may 
always  be  represented  by  the  transverse  shear,  located 
as  the  resultant  of  all  the  external  forces  producing  the 
bending  moment,  multiplied  by  its  lever-arm.  This  is  a 
simple  but  important  matter  of  observation. 

In  the  section  JF  let  the  two  equal  and  opposite 
forces  S  and  —5,  numerically  equal,  act  in  opposite  direc- 
tions; they , will  not,  therefore,  affect-  the  equilibrium  of 
the  beam  or  any  portion  of  it  in  any  way  whatever.  As 
far  as  the  equilibrium  of  the  portion  of  the  beam-yF 
is  concerned,  the  loads  and  the  reactions  may  be  supposed 
to  be  displaced  by  the  couple  5,  —5,  with  the  lever-arm  e, 
and  the  shear  5  acting  upward  in  the  section  JF.  The 
importance  of  this  particular  feature  of  the  analysis  con- 
sists in  showing  that  in  every  bent  beam  carrying  loads 
the  action  of  the  external  forces  (including  the  reaction) 
producing  the  bending  is  equivalent  to  a  couple  whose 
moment  is  Se  acting  about  the  neutral  axis  of  the  section 
>and  the  total  transverse  shear  5  acting  in  the  section. 
The  shear  5  evidently  tends  to  move  or  slide  one  portion 
of  the  beam  past  the  other,  and  an  essential  part  of  the 
operation  of  designing  beams  and  trusses  is  its  determina- 
tion at  various  sections  with  correspondingly  various 
positions  of  loading. 

As  is  well  known,  the  analytical  condition  for  a  maxi- 
mum or  minimum  bending  moment  in  a  beam  is 


Art.  16.]       EXTERNAL  BENDING  MOMENTS  AND  SHEARS.  67 

dM 


dx 


(4) 


From  eqs.  (3)  and  (4)  is  to  be  deduced  the  following 
principle :  The  greatest  or  least  bending  moment  in  any  beam 
is  to  be  found  in  that  section  for  which  the  shear  is  zero. 

The  greatest  bending  moment  obviously  is  the  only 
one  of  importance  in  the  design  of  beams  and  trusses,  and 
eq.  (4)  shows  that  the  section  in  which  it  will  be  found 
can  be  located  by  simple  inspection  of  the  loading.  It  is 
only  necessary  to  sum  up  the  reaction  at  one  end  and  the 
loads  adjacent  to  it,  until  the  point  is  reached  where  the 
summation  is  zero.  This  point  will  usually  be  found 
where  a  load  is  supported.  In  that  case  the  single  load 
may  arbitrarily  be  divided  into  two  parts,  supposed  to  act 
indefinitely  near  to  each  other,  so  that  one  of  the  parts 
may  be  just  sufficient  to  make  the  zero  summation  desired. 
A  single  practical  operation  will  make  this  feature  per- 
fectly clear  and  simple. 

If  the  loading  is  uniformly  continuous  and  of  the 
intensity  p,  in  each  of  the  equations  (i),  (2),  and  (3) 
pdx  is  to  be  used  for  each  of  the  separate  loads  Wv  Wv  Wv 
etc.  The  bending  moment  thus  becomes 

M  - R'x'  -  IWx  - R'y?  -/**•  pdx  =  R'x'  -  \px\      (5) 
The  expression,  for  the  shear  then  becomes 

f  -S-K-**.     . (6) 

A  second  differentiation  gives 


68 


FLEXURE. 


[Ch.  II. 


Or,  the  second  differential  coefficient  of  the  moment 
considered  a  function  of  x  is  equal  to  the  intensity  of  the 
continuous  load. 

This  method  of  passing  from  formulae  for  concentrated 
loads  to  those  for  continuous  loads  is  perfectly  simple  and 
frequently  employed. 


Art.  17. — Intermediate  and  End  Shears. 

The  beam  shown  in  Fig.  i  is  supposed  to  carry  any 
loading  whatever,  and  the  figure  is  consequently  intended 
to  exhibit  a  uniform  load  in  addition  to  a  load  of  con- 
centrations. Inasmuch  as  all  beams  and  other  similar 
pieces  have  considerable  weight,  and  sometimes  great 
weight,  ordinarily  considered  uniformly  distributed  over 
the  span,  this  condition  of  loading  is  that  which  exists  in 
all  actual  cases.  The  amount  of  uniform  loading  per 
linear  unit,  usually  a  foot,  is  represented  by  p,  while  the 


W 


FIG.  i. 

concentrations,  as  heretofore,  are  represented  by  Wlt  W2J 
etc. 

The  determination  of  the  transverse  shear  at  any  sec- 
tion of  a  beam  or  truss  is  one  of  the  most  frequent  as  well 
as  one  of  the  most  important  computations  required  in 
the  design  of  structures.  As  has  already  been  indicated, 
it  is  an  extremely  simple  computation.  It  is  first  neces- 
sary, after  knowing  the  position  of  the  loading,  to  find  the 
reactions  at  both  ends  of  the  span.  In  Fig.  i  the  various 


Art.  17.]  INTERMEDIATE  AND  END  SHEARS.  69 

weights  or  loads  are  separated  by  the  distances  shown,  a' 
being  the  distance  from  \\\  to  the  reaction  R  or  end  of  the 
span.  irc  is  supposed  to  rest  at  the  right  end  of  the  span 
for  a  purpose  that  will  presently  appear.  The  reaction 
R"  at  the  left  end  of  the  span  (not  shown)  resulting  from 
the  concentrated  loads  only  will  have  the  following  value: 


.     (x) 

The  reaction  R"r  at  t'ie  other  end  of  the  span  (not 
shown)  can  be  expressed  by  a  similar  equation,  but  it  is 
simpler  and  more  direct  to  write  it  as  follows  : 


Obviously  the  sum  of  the  two  reactions  R"  and  R"' 
must  be  equal  to  the  total  concentrated  loading. 

That  part  of  the  reaction  due  to  the  uniform  load  ex- 
tending over  the  span  /  will  clearly  be  one  half  of  that 
load  or 


(3) 


The  reaction  R^  is  supposed  to  be  found  at  the  left 
end  of  the  span  and  R,,  at  the  right  end.  The  total  re- 
actions then  will  be  as  follows.  At  left  end  of  the  span  : 


(4) 

At  right  end  of  the  span  : 

.          .  (5) 


/o  FLEXURE.  [Ch.  II. 

The  transverse  shear  at  any  intermediate  section  of 
the  beam  whatever  may  now  readily.be  written.  Let  the 
section  AB  at  the  distance  x  from  the  left  end  of  the  span 
first  be  considered.  The  total  loading  between  that  sec- 
tion and  the  end  of  the  span  is  Wl  +  W2  +  px,  and  it  acts 
downward.  As  the  reaction  R  acts  upward  the  expression 
for  the  shear  will  be 


(6) 


In  this  case  the  section  considered  has  been  taken 
between  two  weights;  let  the  section  at  the  weight  W» 
be  considered,  that  weight  being  at  the  distance  x'  from 
the  end  of  the  span.  The  amount  of  uniform  load  over 
the  length  x'  is  simply  pxf  ,  but  inasmuch  as  the  weight  W3 
is  located  at  the  section  under  consideration,  the  portion 
of  that  weight  which  may  be  taken  as  resting  on  the  left 
of  the  section  considered  is  indeterminate.  In  such  cases 
it  is  proper  and  customary  to  take  any  portion  or  all  of 
the  weight  as  resting  on  either  side  of  the  section,  but 
indefinitely  near  to  it.  If  it  is  a  case  where  the  maximum 
shear  is  desired,  the  single  weight  should  be  taken  in  such 
a  position  as  to  make  the  transverse  shear  as  great  as 
possible.  If  the  case  is  one  where  it  is  desired  to  find  the 
section  at  which  the  total  load  from  that  section  to  the 
end  of  the  span  is  equal  to  the  reaction,  any  portion  may 
be  taken  which  is  found  necessary  to  make  the  equality. 
If,  for  instance,  pxf  +  W1  +  W2  is  less  than  R  while  px'  + 
Wl  +  W2  +  W3  is  greater  than  R,  then  that  portion  of  W3 
which  would  be  considered  on  the  left  of  the  section  but 
indefinitely  near  to  it  would  be  R  —  pxf  —  W1  —  VV2.  The 
remaining  portion  of  W3  would  be  considered  as  resting 
at  the  right  of  the  section  but  indefinitely  near  to  it.  In 
such  a  case  the  transverse  shear  is  zero  at  the  weight  W3. 


Art.  17.]  INTERMEDIATE  AND  END  SHEARS.  7' 

Again,  let  it  be  desired  to  find  the  greatest  upward 
shear  at  W3,  it  being  supposed  that  R  is  greater  than  the 
total  load  between  W3  and  the  left  end  of  the  span.  In 
this  case  no  portion  of  W3  would  be  considered  as  acting 
to  the  left  of  the  section,  but  the  expression  for  the  shear 
would  be 

S^R-px'-Wt-W,..     ....     (7) 

It  can  be  seen  from  the  preceding  statements  that  the 
maximum  transverse  shear  in  the  beam  will  occur  at  the 
ends  of  the  span  where  the  value  of  the  shear  is  the  end 
reaction.  Inasmuch  as  the  end  reaction  R  or  R'  is  thus 
the  greatest  shear  in  the  entire  span,  it  is  a  most  important 
quantity  to  determine  in  the  design  of  beams  and  trusses; 
it  is  the  most  important  single  factor  in  the  determina- 
tion of  the  amount  of  material  required  at  the  end  sections 
of  both  beams  and  trusses.  The  value  of  this  end  shear 
is  given  by  the  values  for  R  and  R'  in  eqs.  (4)  and  (5). 

Since  the  total  transverse  shear  in  any  section  of  a 
beam  is  simply  the  summation  of  all  the  external  loads, 
including  the  reactions  from  one  end  of  the  span  up  to  the 
section  considered,  it  is  evident,  first,  that  that  summation 
may  be  made  from  either  end  of  the  span,  and  second, 
that  the  amounts  so  found  will  be  equal  numerically  but 
affected  by  opposite  signs.  In  determining  the  shear, 
therefore,  in  any  given  case,  it  is  usual  to  make  the  sum- 
mation from  that  end  of  the  span  which  can  be  used  with 
the  greatest  convenience  in  computation. 

Fig.  2  exhibits  a  graphical  representation  of  the  pre- 
ceding treatment  of  intermediate  and  end  shears,  MN 
being  the  length  of  span  shown  in  Fig.  i.  MF  is  the 
reaction  R  laid  off  at  a  convenient  scale.  The  weights  or 
loads  Wv  W2,  V73,  etc.,  are  laid  off  vertically  downward  in 
their  proper  locations  at  the  same  scale,  as  shown.  The 


FLEXURE. 


[Ch.  II. 


vertical  distance  of  G  below  F  is  the  amount  of  uniform 
load  pa'  between  R  and  W\  in  Fig.  i ,  also  laid  down  by  the 
same  scale.  GG\  is,  therefore,  the  shear  in  the  beam  of 


-I-R 


-R' 


FIG.  2. 


Fig.  i  immediately  to  the  left  of  Wv  and  H1Gl  is  the  shear 
immediately  to  the  right  of  the  same  load.  Similarly, 
HLH  being  drawn  horizontally,  HK  is  the  amount  of  uniform 
loading  pa  between  Wl  and  W2.  The  remainder  of  the 
diagram  is  drawn  in  the  same  manner. 

Any  vertical  ordinate  drawn  from  MN  either  up  or 
down  to  the  broken  line  FGHtK  ...  0  represents  the  shear 
at  the  corresponding  point  in  the  span  at  the  same  scale 
used  in  laying  off  the  reactions  and  loads.  QQ1  is  the  shear 
at  the  point  or  section  of  beam  at  Qv  while  TTL  is  the 
shear  at  the  section  T.  The  shear  is  zero  at  W3  where  it 
changes  its  sign.  At  that  point  also  will  be  found  the 
greatest  bending  moment  in  the  beam. 

As  the  diagram  is  drawn  the  shears  on  the  left  of  W3 
and  above  MN  are  positive,  those  on  the  right  of  W3  and 
below  MN  being  negative ;  but  the  diagram  might  have 


Art.  17.] 


INTERMEDIATE   AND  END  SHEARS. 


73 


been  drawn  with  equal  propriety  so  as  to  have  made  R' 
and  the  shears  between  it  and  Ws  positive  and  those  be- 
tween that  load  and  R  negative. 

A  glance  at  the  diagram  shows  that  the  end  shears, 
equal  to  the  reactions,  are  the  greatest  in  the  span. 


+R 

M 

Wl 

W2 

**3                                                     N 

y, 
I 

I 

M 

W4 

-R' 
Uo 

w. 

FIG.  3. 

If  a  beam  carries  a  load  of  concentrations  only  its  shear 
diagram  will  be  illustrated  by  Fig.  3,  in  which  there  are 
five  loads,  the  diagram  being  composed  of  rectangles  only. 
If,  again,  the  load  is  wholly  uniform  Fig.  4  will  represent 
the  shear  diagram  composed  of  two  triangles  with  their 
apices  at  Ct  the  centre  of  the  span  and  point  of  no  shear. 
Any  vertical  ordinate  drawn  from  MN  in  either  figure 


FIG.  4. 

to  the  stepped  line  in  the  one  case  and  to  the  straight  line 
in  the  other  will  represent  the  shear  at  the  section  of  beam 
from  which  the  ordinate  is  drawn.  Those  diagrams  repre- 


74 


FLEXURE. 


[Ch.  II. 


sent  completely  the  graphical  treatment  of  shears  in  all 
cases. 


Art.  18.— Maximum  Reactions  for  Bridge  Floor  Beams. 

Three  transverse  floor  beams  of  a  railroad  bridge  are 
represented  in  Fig.  i  separated  by  the  two  spans  /t  and  / 
which,  in  a  bridge,  represent  the  panel  lengths.  The 
members  AB  and  BC  supporting  the  weights  Wlt  W2, 
etc.,  indicate  the  stringers  which  carry  the  railroad  track 
and  the  train.  The  two  beams  or  stringers  AB  and  BC 
are  considered  simple  non-continuous  beams  resting  on 
the  floor  beams,  but  not  necessarily  nor  usually  on  their 
tops.  The  problem  is  to  determine  the  position  of  the 
locomotive  or  other  train  loads  on  the  adjacent  two  short 
opans  /t  and  /,  so  that  the  reaction  R  on  the  floor  beam 
between  shall  have  its  greatest  value. 

In  Fig.  i  let  a  section  of  the  beam  be  shown  at  R,  and 
let  %  and  %\  be  measured  from  the  right  ends  of  the  two 
spans  as  shown  in  Fig.  i,  while  Wi,  Wz,  .  .  .  W±  repre- 


j 

*                                                                1 

R'         w,          w?L  Xl  * 

W3             W4                  W5 

A 

Q  |*)  —  -a  -  —  (ffj~  ~k 

-  "(^r/"  -Cr  -  ~\^j  '  ~  ~d-  -  -  -  \jy  '  ~x~  '  "   i 

i                              ^ 

B 

5 

-  -f-2    • 

FIG.  i. 


sent  a  train  of  weights  or  wheel  concentrations  passing  over 
the  two  spans  from  right  to  left.  If  R'  and  R  are  the 
reactions  at  A  and  B,  respectively: 


.  (i) 


Art.  18.]     MAXIMUM  REACTIONS  FOR  BRIDGE  FLOOR  BEAMS.          75 

Then  if  the  moments  of  weights  and  reactions  be  taken 
about  C  at  the  right-hand  end  of  span  1%  : 


R'(h  +/2)  -  (Wia  +  (Wi  +  W2)%)  -  (W, 


(2) 


Hence,  since  R'l  \  is  equal  to  the  quantity  within  the 
second  parenthesis  of  the  first  member  of  eq.  (2)  : 


LI  - 

h-irk 

t)d-2  Wx+Rl2=o  ......     (3) 

In  order  that  the  reaction  R  may  have  its  greatest 
value  it  must  remain  unchanged  when  a  small  move- 
ment of  the  train  is  made.  If  therefore  x  +  dx  and  xi+dx 
be  written  for  x  and  %\,  respectively,  in  eq.  (3)  and  if  eq.  (3) 
be  subtracted  from  the  result  so  obtained,  the  following 
equations  will  be  found  : 


=  I  W, 


Eq.  (4)  shows  the  position  of  loading  for  the  greatest 
value  of  the  reaction  R.  It  means  simply  that  the  ratio 
between  the  amount  of  loading  on  span  h  and  the  total 
load  on  both  spans  shall  be  the  same  as  the  ratio  between 
the  span  l\  and  the  sum  of  the  two  spans  (Ii+l2).  Inas- 
much as  the  load  may  move  in  either  direction  12  may 


76  FLEXURE.  [Ch.  II. 

be  written  for  l\  in  the  numerator  of  the  first  member 
of  eq.  (4). 

Clearly  the  two  weights  W\  and  Wz  in  the  preceding 
equations  represent  all  the  loads  resting  on  span  l\  whether 
there  be  two  such  weights  or  any  number  whatever.  Sim- 
ilarly the  weights  indicated  by  the  summation  sign  in  the 
second  member  of  eq.  (4)  represent  the  total  load  on  both 
spans.  If  /i  =£2,  as  is  usually  the  case,  the  first  member 
of  eq.  (4)  has  the  value  of  one-half. 

As  in  all  such  cases  there  may  be  more  than  one  posi- 
tion of  the  loading  which  will  satisfy  the  criterion  eq.  (4) ; 
in  that  case  it  is  necessary  to  determine  which  of  those 
conditions  will  give  the  maximum  of  the  "  greatest  values  " 
ofR. 

Inasmuch  as  the  sum  of  the  weights  on  the  span  h 
does  not  change  for  any  value  of  %\  equal  to  or  less  than 
b,  it  follows  that  a  weight  may  be  taken  at  the  point  of 
support  B  in  satisfying  eq.  (4).  This  will  simplify  the  use 
of  eq.  (3)  in  writing  the  expression  for  R.  If  x\  =b  there 
may  at  once  be  written  from  eq.  (3) : 

L  fc+** 

-(Wia+(Wi  +  W*Wf+(Wi+W*  +  Wi)c+(Wi  +  .  .  .  +W,}d-\-  2  Wx 

*—  -g-  --(5) 

This  equation  gives  the  value  of  R  desired,  and  it  is 
so  written  that  numerical  values  may  readily  be  computed 
by  the  use  of  tables.  If  /i=/2,  as  is  usual,  the  ratio  of 
those  two  quantities  becomes  unity. 

Art.  19. — Greatest  Bending  Moment  Produced  by  Two 
Equal  Weights. 

Fig.  i  represents  a  non- continuous  beam  with  the  span  / 
supporting  two  equal  weights  P,  P.  These  two  weights  or 
loads  are  to  be  kept  at  a  constant  distance  apart  denoted 
by  a. 


Art.  19.]    BENDING  MOMENT  PRODUCED  BY  TWO   WEIGHTS.          77 

It  is  required  to  find  that  position  of  the  two  loads 
which  will  cause  the  greatest  bending  moment  to  exist 
in  the  beam,  and  the  value  of  that  moment.  The  reac- 
tion R  is  to  be  found  by  the  simple  principle  of  the  lever. 
Its  value  will  therefore  be 


(i) 


Since  the  reaction  R  can  never  be  equal  to  2P,  IP, 
or  the  shear,  must  be  equal  to  zero  at  the  point  of  applica- 
tion of  one  of  the  loads  P.  In  searching  for  the  greatest 


t~::::%. 
jit 

m 


FIG. 


moment,  then,  it  will  only  be  necessary  to  find  the  moment 
about  the  point  of  application  of  one  of  the  forces  P.     It 
will  be  most  convenient  to  take  that  one  nearest  R. 
The  moment  desired  will  be 


x      ax 


dM_ 
dx  ~°  ~2    \ 

1  a 

.'.  x= . 

2  4 

This  value  in  eq.  (2)  gives 

' "  "x>  .      .  ( ) 


78  FLEXURE.  [Ch.  II. 

Since 

d*M _  _4P 
dx2  =     "  I  ' 

it  appears  that  M1  is  a  maximum. 

The  shear  5  in  the  section  RP  of  the  span  will  be   the 
reaction  R  as  given  by  eq.  (i) : 


Throughout  the  section  a  the  shear  S'  will  be 

2Pi       as 


(5) 


Finally,    between    the    right    abutment    and    the    nearest 
weight  the  shear  5t  will  be 

oP/          n\ 

....     (6) 

If  the  separating  distance,  a,  between  the  two  weights 
be  increased  a  value  may  be  reached  so  great  as  to  make 
the  bending  moment  of  the  pair  of  weights  less  than  that 
produced  by  placing  one  of  them  at  the  centre  of  the  span. 
This  limiting  value  of  a  may  easily  be  found.  The  moment 
at  the  centre  of  span  produced  by  placing  a  single  weight 
P  there  is 

2*2         4  " 

By  using  eq.  (3) 

M'=M,;    /  .^-IV/.E)'.  (7) 


Art.  20.]    BENDING  MOMENTS  OF  CONCENTRATED  LOADS.  79 

By  solving  this  equation 

(8) 


Whenever,  therefore,  the  separating  distance  a  is  equal 
to  or  greater  than  .586  span  length,  the  moment  should 
be  found  by  placing  a  single  weight  P  at  the  centre  of  the 
span. 

Art.  20.  —  Position  of  Uniforc?  Load  for  Greatest  Shear  and 
Greatest  Bending  Moment  at  any  Section  of  a  Non- 
Continuous  Beam  —  Bending  Moments  of  Concentrated 
Loads. 

A  continuous  load  of  -uniform  density  is  frequently 
employed  in  structural  operations  both  for  beams  and 
trusses,  and  it  is  essential  to  place  such  a  load  so  as  to 
produce  the  greatest  effect  both  for  shears  and  moments. 
The  position  of  loading  for  the  greatest  shear  will  first  be 

found. 

A  continuous  train  of  any  given  uniform  density  ad- 
vances along  a  simple  beam  of  span  I.  It  is  required  to 
determine  what  position  of  loading  will  give  the  greatest  shear 
at  any  specified  section. 

In  Fig.  i,  CD  is  the  span  /,  and  A  is  any  section  for 


C                                            A            B 

D 

1 

^^jj$ 

FIG.  i. 

which  it  is  required  to  find  the  position  of  the  load  for  the 
greatest  transverse  shear.  The  load  is  supposed  to  ad- 
vance continuously  from  C  to  any  point  B.  Let  R  be  the 


'8o  FLEXURE.  [Ch.  II. 

reaction  at  D,  and  IP  the  load  between  A  and  £.     The 
shear  5'  at  A  will  be 

R-IP=Sf  .....     ,     .     (i) 

Let  Rf  be  that  part  of  R  which  is  due  to  IP,  and  R" 
that  part  due  to  the  load  on  CA  ;  evidently  R'  is  less  than 
IP.  Then 


If  A  B  carries  no  load,  R'  and  IP  disappear  in  the  value 
of  5.     Hence 


is  the  shear  for  the  head  of  the  train  at  A.  S  is 
greater  than  5'  because  IP  is  greater  than  R'  .  But  no  load 
can  be  taken  from  AC  without  decreasing  R'  '.  Hence  the 
greatest  shear  at  any  section  will  exist  when  the  load  extends 
from  the  end  of  the  span  to  that  section,  whatever  be  the  den- 
sity of  the  load. 

In  general,  the  section  will  divide  the  span  into  two  un- 
equal segments.  The  load  also  may  approach  from  either 
direction.  The  greater  or  smaller  segment,  then,  may  be 
covered,  and,  according  to  the  principle  just  established, 
either  one  of  these  conditions  will  give  a  maximum  shear. 
A  consideration  of  these  conditions  of  loading  in  connec- 
tion with  Fig.  i  ,  however,  will  show  that  these  greatest 
shears  will  act  in  opposite  directions. 

When  the  load  covers  the  greater  segment  the  shear  is 
called  a  main  shear  ;  when  it  covers  the  smaller,  it  is  called 
a  counter  shear. 

The  determination  of  the  greatest  bending  moment 
at  any  section  A  of  a  beam  or  truss,  exemplified  by  Fig.  i, 
traversed  by  a  continuous  train  of  uniform  density  is  a 
very  simple  matter.  It  is  clear  that  every  part  of  the 


Art.  20.]       BENDING  MOMENTS  OF  CONCENTRATED  LOADS.  81 

uniform  load  resting  on  the  beam  will  produce  bending  at 
any  section  considered  ;  and  it  is  further  obvious  that  every 
part  of  that  uniform  loading  will  create  a  bending  moment 
at  A  of  the  same  sign.  It  follows,  therefore,  that  the 
entire  span  should  be  covered  by  the  uniform  train  in  order 
to  produce  a  maximum  bending  moment  at  any  section 
of  the  beam  or  truss,  and  that  this  single  position  of  the 
train  will  give  the  maximum  bending  moment  throughout 
the  entire  span. 

The  preceding  position  of  moving  load  is  taken  only  for 
a  train  of  uniform  density  or  for  a  series  of  uniform  con- 
centrations, each  pair  of  which  is  separated  by  the  same 
distance  as  every  other  pair,  i.e.,  for  a  uniformly  distributed 
system  of  uniform  concentrations.. 

The  general  case  of  a  simple  beam  loaded  with  any 
system  of  weights  may  be  represented  by  Fig.  2,  in  which 
the  beam  carries  three  loads  Wv  W2,  and  Ws,  spaced  as 
shown.  The  reactions  or  supporting  forces  R  and  Rf  are 
determined  in  the  usual  manner  by  the  law  of  the  lever. 
Hence 


A  similar  value  may  be  written  for  Rf,  but  it  is  simpler 
after  having  found  one  reaction  to  write 

Rf=W,  +  W2  +  W3-R  .......     (3) 


The  beam  itself  being  supposed  to  have  no  weight,  the 
bending  moments  at  the  points  of  application  of  the  loads 
will  be 


•     •     (4) 


82 


FLEXURE. 


[Ch.  II. 


After  substituting  the  value  of  R  from  eq.  (2)  in 
eqs.  (4)  the  moments  in  the  latter  equations  will  be  com- 
pletely known. 


S=R 


S'=-R' 


FIG.  2. 

The  bending  moment  produced  by  each  weight  will  be 
represented  by  the  ordinates  of  the  triangles  shown  in  Fig.  2, 
the  resultant  moments  at  the  points  of  application  of  the 
weights  being  given  by  eqs.  (4).  The  ordinate  CD  repre- 
sents Mj  in  eqs.  (4)  by  any  convenient  scale.  Similarly 
FH  represents  M2  in  eqs.  (4),  and  KL,  M3.  The  lines 
AC,  CF,  FK,  and  KB  are  then  drawn.  Any  vertical  inter- 
cept between  AB  and  the  polygon  ACFKB,  found  in  the 
manner  explained,  will  represent  the  bending  moment 
at  the  point  where  the  intercept  is  drawn,  and  to  the  scale 
at  which  Mv  M2,  and  M3  are  laid  down.  This  intercept  is 
simply  the  sum  of  the  intercepts  of  the  triangles,  each 
representing  the  partial  bending  moment  due  to  a  single 
weight. 


Art.  21.]   BENDING  MOMENT  IN  A   NON-CONTINUOUS   BEAM.         83 

Obviously  the  bending  moments  of  any  number  of  loads 
of  any  magnitude  or  of  a  uniform  load,  even,  may  be  treated 
or  represented  in  the  same  manner. 

The  lower  portion  of  Fig.  2  is  the  shear  diagram  drawn 
precisely  as  explained  for  Fig.  3  of  Art.  17. 

Art.  21.— Greatest  Bending  Moment  in  a  Non-Continuous 
Beam  Produced  by  Concentrated  Loads. 

The  position  of  the  moving  load  for  the  greatest  bend- 
ing moment  at  any  section  of  a  non -continuous  beam  may 
be  very  simply  determined.  In  Fig.  i,  let  FG  represent 
any  such  beam  of  the  span  /,  and  let  any  moving  load  what- 
ever, as  V/i  .  .  .  W  nr  -  -  -  Wn  advance  from  F  toward  G. 
Let  C  be  the  section  at  which  it  is  desired  to  determine  the 
maximum  bending  moment,  and  let  n'  loads  rest  to  the 
left  of  C,  while  n  is  the  total  number  of  loads  on  the  span. 
Finally,  let  x'  represent  the  distance  of  W ' n>  from  C  and  to 
the  left  of  that  point,  while  x  is  the  distance  of  Wn  to  the 
left  of  F.  If  a  is  the  distance  between  W1  and  W2,  b  the 
distance  between  W.,  and  W3,  c  the  distance  between  W3 
and  W4,  etc.,  the  reaction  R  at  G  will  be 


W'i 

W.- 


The  bending  moment  M  about  C  will  then  take  the 
value 


84 


FLEXURE. 


[Ch.  II. 


Wl(a 
W2( 


Or,  after  inserting  the  value  of  R  from  above, 
M  =   [W,a  +  (W,  +  W2)b 


-  W,a  -  (W1  +  W2)b  -  (W,  +  W2  +  W3)c 


If  the  moving  load  advances  by  the  amount  Ax,  the 
moment  becomes,  since  Ax  =  Ax'  , 


o  O 


-l'- 


C.G. 

FIG.  i. 


-(W,+W2+    ...    +Wn*)4*.        (3) 

Hence,  for  a  maximum,  the  following  value  must  never 
be  negative  : 

Mf-M==Aoc\^(W,  +  W2  +  Wz+  .  ..  +Wn) 

-(Wi  +  W2+  ...  +Wn.)\  =o.     (4) 
Or  the  desired  condition  for  a  maximum  takes  the  form 


.      ...     (5) 


Art.  21.]     BENDING   MOMENT  IN  A  NON-CONTINUOUS  BEAM.        85 

It  will  seldom  or  never  occur  that  this  ratio  will  exactly 
exist  if  Wn'  is  supposed  to  be  a  whole  weight;  hence  Wn> 
will  usually  be  that  part  of  a  whole  weight  at  C  which  is 
necessary  to  be  taken  in  order  that  the  equality  (5)  may 
hold. 

It  is  to  be  observed  that  if  the  moving  load  is  very 
irregular,  so  that  there  is  a  great  and  arbitrary  diversity 
among  the  weights  W,  there  may  be  a  number  of  positions 
of  the  moving  load  which  will  fulfil  eq.  (5),  some  one  of 
which  will  give  a  value  greater  than  any  other;  this  is 
the  absolute  maximum  desired. 

From  what  has  preceded,  it  follows  that  Wnf  may 
always  be  taken  at  the  point  C  in  question;  hence  x'  in 
eq.  (2)  may  always  be  taken  equal  to  zero  when  that 
equation  expresses  the  greatest  value  of  the  moment.  The 
latter  may  then  take  either  of  the  two  following  forms  : 

l  +  W2)b  +  .  .  .  +  (W1  +  W, 


x]  -  W>  -  (W, 
-  ...  -(WL 


(6a) 


In  these  equations  %  corresponds  to  the  position  of 
maximum  bending,  while  the  sign  (?)  represents  the  dis- 
tance between  the  concentrations  Wn>-v  and  Wn>. 

The  preceding  equations  give  the  greatest  bending 
moments  at  any  arbitrarily  assigned  points  in  the  span. 
There  remains  to  be  determined  the  point  at  which  the 
greatest  moment  in  the  entire  span  exists,  and  the  mag- 
nitude of  that  greatest  moment. 


86  FLEXURE.  [Ch.  II. 

It  has  already  been  shown  that  for  any  given  condition 
of  loading  the  greatest  bending  moment  in  the  beam  will 
occur  at  that  section  for  which  the  shear  is  zero.  But  if 
the  shear  is  zero,  the  reaction  R  must  be  equal  to  the  sum 
of  the  weights  (Wi+W2+.  .  .+Wn>)  between  G  and  C, 
the  latter  now  being  the  section  at  which  the  greatest 
moment  in  the  span  exists. 

Hence  for  that  section  eq.  (5)  will  take  the  form 

r  R  (  . 


.     •     .+Wn 

Hence 

R=j(Wi+W2+.  .  .+Wn).      /.     •     •     (8) 

The  relations  existing  in  eqs.  (7)  and  (8)  can  obtain 
only  if  the  centre  of  gravity  CG  in  Fig.  i  is  at  the  dis- 
tance I'  from  F,  showing  that  the  centre  of  gravity  of  the 
load  is  at  the  same  distance  from  one  end  of  the  beam 
as  the  section  or  point  of  greatest  bending  is  from  the 
other.  In  other  words,  the  distance  between  the  point  of 
greatest  bending  for  any  given  system  of  loading  and  the 
centre  of  gravity  of  the  latter  is  bisected  by  the  centre  of  span. 

If  the  load  is  uniform,  therefore,  it  must  cover  the  whole 
span. 

It  is  to  be  observed  that  eq.  (6)  is  composed  of  the  sums 
Wv  Wj  +  Wjj,  etc.,  multiplied  by  the  distances  a,  b,  c,  etc. 
Again,  as  in  the  equation  immediately  preceding  eq.  (2), 
the  expression  for  the  moment,  M,  may  be  taken  as  corn- 
posed  of  the  positive  products  of  each  of  the  single  weights 
Wi,  W2,  etc.,  multiplied  by  its  distance  from  any  point 
distant  x  to  the  right  of  Wn  and  of  the  negative  products 
similarly  taken  in  reference  to  the  section  located  by  x1  ', 
as  shown  by  eq.  (6a). 


I 

^.rt.  21.] 

\  1  II  1  1  I 

1  I  1  1 

1  I  1  1 

1  1  1  1 

III!!!! 

1  1  1  1 

1  1  r  1  1 

m 

45.0                   8 

240 

75.0       .13 

630  > 

105.0      18 

1170  „ 

135.0      23 

1860 

154.5                     32 
2485 

174.0      37 

3205  > 

193.5         43 

4040  j 

<                 ? 
5)       8'     <2 

•345                      ( 

5  5'  ©5'©  5'©        9'       $ 

5            7               8 
^   5'(R)    6'  (? 

18 
17 
16 
15 
14 
13 
12 
11 
10 
Loa 

L0£ 

Mo 
Mo 

.109 

24550 

101 

22910 

so 
19880 

91 

17000 

86 

14"270 

77 

11690 

72 

10190 

426.0                104 

22420 

411.0      96 

20860 

381.0      91 

17980 

351.0      86 

15250 

321.0                      81 

12670    m 

291.0     72 

J  10240 

271.5         67 
8830 

408.5                 99 

20380 

391.5       91 

18900 

361.5      80' 

16170 

331.5      81 

ia590 

301.5                      76 

11160 

271.5      67 
8880 

252.0          02 

7570 

387.0                 93 

18060 

372.0      85 

16670 

342.0      80 

14120 

312.0     75 

11720 

282.0                    70 

9470 

252.0      61 

7370 

232.5        56 

6180 

307.5                 88 

16220 

352.5      80 

14910 

322.5      75 

12500 

292.5     70 

10250 

262.5                    65 

8150 

232.5      50 

6200 

213.0          51 

5110 

348.0                 79 

13090 

333.0      71 

11900 

303.0     00. 

9780 

273.0      61 

7800 

243.0                    56 

5970 

213.0      47 

4290 

193.5          42 

3370 

318.0                 74 

11500 

303.0     60 

10400 

273.0      61 

8410 

243.0      50 

6580 

213.0                     51 

4900 

183.0      42 

3370 

103.5          37 

2555 

238.0                69 

10060 

273.0      61 

jgggQ 

243.0      06 

7200 

213.0      51 

5520 

183.0                    46 

3990 

153.0      37 

2805 

133.5          32 

1885 

258.0                 C4 

8770 

243.0     56 

7810 

213.0      51 

6130 

183.0     40 

4600 

153.0                      41 

3220 

123.0      32 

1992 

103.5          27 

1368 

223.0                 60 

6950 

213.0     43 

6110 

133.0      43 

4670 

153.0      33 

3380 

123.0                      33 

2240 

93.0       24 

1248 

73.5            19 
780 

ds  and  moments  are  for  one  rail 
ids  given  in  thousands  of  pounds 
ments  «    **          «         «  foot  pounds 
ments  are  expressed  to  a  limit  of  error  of  0.1  per  cent 

LEI. 

LLU 

MM  1  II 

III! 

MM 

MM 

ii  ii  i  ii  r 

JIJI 

Mill 

II  II    MM 

50 

520 

253.0                 64 

7740     k 

288.0      69 

9810^ 

318.0      74 

12030 

348.0      7D 

14400^ 

307..5                     88 

16100      > 

387.0      93 

17930^ 

400.5          99 

19850^ 

420.0    104              109 

21900^ 

10-1               11-2       12-3       13-4        U 

^                 lSj-6        16-7          17-8       lSj-9             3  p.l.ft. 

9       9'       ©  5'®)    6'  $T 

•5)  122  s' 

61 

500 

53 

6310 

5514" 

40 

4164 

35 

2965 

30 

1914 

21 

1014 

16 

605 

.10                 5 

292.5      97.5 

50 

340 

213.0                48 

5240 

198.0      40 

4524 

168.0      35 

3325 

138.0     30 

2275 

108.0                     25 

1374 

78.0       10 

624 

58.5           11 

312 

39.0        5 

97.5 

51 

270 

193.5                 43 

4280 

173.5      35 

3632 

148.5      30 

2580 

118.5     25 

1682 

88.5                      20 

932 

58.5        11 

331.5 

39.0            6 

117 

110 

174.0                37 

3230 

159.0      29 

2678 

129.0     24 

1808 

99.0       19 

1088 

69.0                      14 
518 

39.0         5 

97.5 

.ft. 

40 

240 

104.5                32 

2460 

13U.5      24 

1980 

109.5     19 

1260 

79.5        14 

690 

49.5                        9 

270 

31 

550 

135.0                 23 

1245 

120.0      15 

900 

90.0       10 

450 

60.0        5 

150 

20 

227 

105.0                 13 

720 

90.0        10 

450 

00.0        5 

150 

21 

.55 

75.0                  13 

345 

60.0         0 

150 

16 

32 

45.0                   8 

120 

MOMENT  TABLE 

COOPER'S  E-60  LOADING 
Two  213-ton  Engines+6000  Ibs.  p. 
Scale:  1"=15' 

8 

56  

(To  face  page  87.) 


Art.  21.]     BENDING  MOMENT  IN  A   NON-CONTINUOUS   BEAM.      87 

The  practical  application  of  the  preceding  formulae 
can  therefore  best  be  effected  by  means  of  a  tabulation  of 
moments  like  that  shown  in  Table  I,  taken  from  the  stand- 
ard specifications  of  the  N.  Y.  C.  R.  R.  Co.  for  1915.  The 
wheel  weights  and  train  loads  shown  in  the  table  are  for 
one  rail  only,  i.e.,  they  are  half  those  for  one  track.  By 
comparing  the  weights  and  spacings  with  those  in  Fig.  i 
and  eq.  (6)  it  will  be  seen  that  W\  =  15,000  Ibs.;  W^  — 
30,000  Ibs.;  W3=3o,ooo  Ibs.,  etc.,  and  that  a  =  8  ft.;  6  = 
5  ft.;  c  =  s  ft.,  etc. 

The  arrangement  of  Table  I  essentially  as  shown  has 
been  used  for  a  long  time  to  expedite  the  computations  of 
moments  and  shears  produced  by  wheel  concentrations, 
followed  by  a  heavy  uniform  load.  It  will  be  noticed  that 
the  first  line  at  the  top  of  the  diagram  shows  the  progress- 
ive sums  of  the  individual  loads  beginning  at  the  left- 
hand  end,  i.e.,  at  Wi,  in  connection  with  the  progressive 
sums  of  the  distances  between  the  centres  of  each  pair 
of  wheels.  The  second  line  (in  the  larger  figures)  is  the 
progressive  sums  of  the  moments  of  the  wheel  loads  about 
the  centre  of  Wi,  i.e.,  1860  is  the  moment  of  W%,  Wa, 
W±,  and  Ws  about  the  centre  of  W\.  Each  of  the  hori- 
zontal spaces  below  the  heavy  line  on  which  the  wheel 
concentrations  rest  contains  one  line  of  small  figures  and 
one  line  of  large  figures.  The  small  figures  are  the  pro- 
gressive sums  of  the  distances  from  the  head  of  the  uniform 
moving  load  or  from  each  successive  wheel  to  each  of  the 
wheel  weights  in  the  series.  The  larger  figures  give  the 
progressive  sums  of  the  moments  of  the  wheel  weights 
beginning  with  Wig  about  the  head  of  the  uniform  load, 
i.e.,  19.5X5  =97-5>  and  19.5X10+97.5  =292.5.  Each  hori- 
zontal space  is  seen  to  begin  at  the  vertical  heavy  line 
under  each  weight  taken  in  succession  and  to  contain  the 
progressive  sums  of  the  moments,  weights,  and  distances 


88  FLEXURE.  [Ch.  II. 

about  or  from  each  such  weight,  as  is  clear  on  examining 
the  diagram.  At  the  left  of  each  horizontal  line  there  is 
found  the  number  of  the  wheel  load  under  which  the  right- 
hand  end  of  the  line  begins. 

The  diagrammatic  exhibit  of  these  various  numerical 
quantities  will  enable  the  reactions,  shears,  and  greatest 
moments  at  any  point  in  the  span  to  be  readily  deter- 
mined. 

When  a  uniform  train  load  is  a  part  of  the  system 
of  loading  it  is  only  necessary  to  consider  any  section 
of  it  as  acting  through  its  centre  of  gravity,  i.e.,  through 
its  mid-point.  Taking  that  centre  as  its  point  of  appli- 
cation the  separating  space  is  the  distance  from  that  point 
to  the  nearest  concentration.  If  in  Table  II  20  ft.  of 
train  load  be  used,  that  train  weight  will  be  60,000  Ibs. 
applied  at  the  distance  10  +  5  =  15  ft.  from  load  18.  This 
simple  operation  is  all  that  is  needed  for  any  uniform 
load  or  for  a  series  of  sections  of  uniform  load. 

Table  II  is  a  table  of  maximum  moments,  end  shears* 
and  floor-beam  reactions  for  girders  having  spans  up  to  125 
ft.,  and  it  is  taken  from  the  New  York- Central  Railroad 
Specifications  for  1915.  The  shears  and  floor-beam  reactions, 
like  the  results  shown  in  Table  I,  are  given  in  thousands 
of  pounds  and  are  for  one  rail  only.  The  moments  are  given 
in  thousands  of  foot-pounds,  like  the  moments  shown  in 
Table  I.  The  loading  is  the  same  as  that  shown  by  the 
diagram  in  Table  I,  except  that  the  results  for  spans  up 
to  a  maximum  of  n  ft.  are  found  by  using  a  special 
loading  of  two  72,ooo-lb.  axle  loads  7  ft.  apart,  or  36,000 
Ibs.  for  each  rail.  The  maximum  moments  are  found  for 
the  conditions  of  loading  given  by  the  criterion,  eq.  (5), 
of  this  article.  The  maximum  floor-beam  reactions  are 
found  by  eq.  (5)  of  Art.  18,  in  accordance  with  the 
criterion,  eq.  (4),  of  the  same  article. 


Art.  2i.l     BENDING  MOMENT  IN  A  NON-CONTINUOUS  BEAM.        89 


TABLE  II. 

TABLE   OF   MAXIMUM    MOMENTS,    END   SHEARS  AND   FLOOR- 
BEAM   REACTIONS   FOR   GIRDERS. 

Moments  in  Thousands  of  Foot-pounds. 
Shears  and  Floor-beam  Reactions  in  Thousands  of  Pounds. 

Loading- Two  E  60  Engines  and  Train  Load  of  6000  Ibs.  per  Foot  or  Special 
Loading  Two  72,ooo-lb.  Axle  Loads  7  Ft.  C  to  C. 

Results  for  One  Rail.     Results  from  Special  Loading  Marked  *. 


Span. 
Ft. 

Maximum 
Moments. 

End 
Shear. 

Floor- 
beam 
Reaction. 

Span. 

Maximum 
Moments. 

End 
Shear. 

Floor- 
beam 
Reaction. 

6 

8 
9 

10 

ii 

12 

13 
14 

*  45-0 
*  54-0 
*  63.0 
*  72.0 
*  81.0 

*  90.0 
*  99.0 

I2O.O 
142.5 
165.0 

*36.o 
*36.o 
38-6 
41-3 
*44-0 

*46.8 
49.1 
52-5 
55-4 
57-8 

*36.o 
40.0 
47-i 
52-5 
56-7 

60.0 

65-5 
70.0 

73-9 

78.2 

35 
36 
37 
38 
39 

40 

4i 
42 

43 
44 

784.5 
823.0 
861.6 
900.0 
940.0 

983-4 
IO27.0 
1070.4 
III3-9 
1157.4 

103.8 
105.9 
107.8 
109.7 
III.4 

II3.I 
II5.2 
II7.2 
II9.0 
120.8 

146.4 

149-3 
152.2 
155-6 
158.8 

I62.O 

15 

187.5 

60.0 

82.0 

45 

1201.  I 

122.5 

16 

210.0 

63.8 

85-3 

46 

1244.4 

124.2 

17 

232.5 

67.1 

88.2 

47 

1287.9 

125.0 

18 
19 

20 

255-0 
280.0 

309.5 

70.0 

72.6 

75-0 

91.0 
94-3 

98-3 

48 
49 

50 

I33I-4 
1378.3 

1426.3 

127.5 

129.2 
130.8 



21 

22 
23 

339-0 
368.5 
398.2 

77.1 
79.1 
80.9 

101  .9 
105.2 
108.2 

5i 

52 
53 

1474-7 
1522.8 
I57I.O 

132.5 
134.1 

135.  7 

24 

427.8 

83.1 

110.9 

54 

1622.2 

137.4 

25 

457-5 

85.2 

113.5 

55 

1675.2 

1  10   o 

26 

27 

487.2 
516.9 

87.1 
88.9 

116.6 

120.  I 

56 

57 

1728.6 
I78l  .9 

140.6 
142.2 



28 

548.3 

90.6 

123.4 

58 

1835.1 

143.8 

29 

582.0 

92.3 

126.5 

59 

1891.4 

145.4 

30 

615.8 

94-6 

129.4 

60 

1949.4 

147.0 

31 

649.3 

96.6 

132  .  7 

61 

2OO7  .  5 

148  6 

32 

33 
34 

683.2 
716.9 
750.6 

98.6 
100.4 

102.  I 

136.5 
140.0 
143.2 

62 

63 
64 

2065.4 
2123.4 
2183.3 

150.2 
152.0 
153.8 



9° 


FLEXURE. 

TABLE  II. — (Con.) 


[Ch.  II. 


Span. 

Maximum 
Moments. 

End 
Shear. 

Floor 
beam 
Reaction. 

Span. 

Maximum 
Moments. 

End 
Shear. 

'     Floor- 
beam 
Reaction. 

65 

2246    ^ 

155-7 

95 

4408  .  4 

215.4 

66 

2^OQ    ^ 

I  S7    S 

96 

449O   7 

217    2 

67 

68 

2372.3 

2435  .  4 

159.6 
161.7 

97 

98 

4573-5 
4659.8 

219.2 

221  .2 

60 

2408  .  4 

163.8 

99 

4743.8 

22^    I 

7O 

2560.4 

165.8 

IOO 

4830  .  o 

225.O 

71 

2624   5 

l6?    7 

IOI 

49l6  9 

226  8 

72 

2688.3 

170.0 

102 

5004  .  o 

228.6 

7-1 

2750.9 

172.2 

103 

5115.5 

230.4 

74 

28lQ  4 

174  4 

IO4 

5212.8 

2^2    ^ 

7c 

2888.6 

176.  s 

IO5 

5306  .  5 

2^4.  I 

76 

77 

2958.0 
3028  6 

178.6 
180  6 



1  06 
IO7 

5401  .  3 

S499   2 

235-9 

2^7    7 

78 

3096.6 

182.5 

1  08 

5617.0 

239.4 

7Q 

3l68    2 

184  4 

109 

5727.6 

241    2 

80 

^240  7 

186  3 

no 

S829.6 

24^    O 

81 

^11  .4 

188  4 

III 

5937-4 

244.8 

82 

^85.1 

190.4 

112 

6040  .  o 

246.6 

81 

3459.6 

192.^ 

113 

6148.2 

248.3 

"O 

84. 

3534-6 

194.2 

114 

6258.0 

25O.O 

85 

^610  4 

1  06  I 

IIS 

6366  8 

251   8 

86 

^689  .  4 

IQ8    I 

116 

6478.0 

25^  6 

87 

^766  5 

200  i 

117 

6586.1 

255.  ^ 

88 

1846  .  o 

202.  1 

118 

6696  .  6 

257.0 

80 

^924  ^ 

204.0 

119 

6808.3 

258.8 

QO 

4005  8 

205.8 

I2O 

6921.6 

260.5 

QI 

4O84   4 

2O7    7 

121 

70^0.  s 

262    2 

Q2 

4164  o 

2OQ   7 

122 

7143.8 

264.0 

Q-I 

4246  6 

21  1    6 

12^ 

7260.  I 

265.7 

" 

94 

4328.0 

213-5 

124- 
125 

7376.4 
7495.2 

267.4 
269    I 

PROBLEM. 

Let  a  single-track  railroad  plate  girder  with  an  effective 
span  of  88  ft.  be  traversed  from  right  to  left  by  the 
moving  load  shown  in  Table  I.  It  is  required  to  find 
the  greatest  bending  moments  and  shears  at  the  centre 


Art.  21.]      BENDING  MOMENT  IN  A  NON-CONTINUOUS  BEAM.        91 

and  quarter-points  of  the  span,  the  dead  load  or  own 
weight  of  the  girder,  floor  system  and  track  being  taken 
at  1800  Ibs.  per  linear  foot. 

Dead  Load. 

By  eq.  (6)  of  Art.  22  the  bending  moments  at  the 
quarter-point  and  centre  are,  since  the  reaction  R  is 
44  X  900  =39,600  Ibs. ; 

Quarter-point.  Centre. 

X  =  \l  =  22  ft.  X  =  %l  =  44  ft. 

M=£(lx-x2) .  .  .654,000  ft.-lbs.        871,000  ft.-lbs. 

2 

By  eq.  (7)  of  Art.  22,  the  shears  at  end,  quarter- 
point,  and  centre  are : 

End.  Quarter-point.  Centre. 

X=o  #  =  22  ft.  #=44  ft. 

Shear  =  39, 600  Ibs.       19, 800  Ibs.  zero 

Moving  Load. 

If  weight   W4  be  placed  at  the  quarter-point  of  the 

span,   14  wheel  weights  will  rest  on  the  girder  with  W\± 

I' 
5  ft.    from   the   right-hand   end    of   the   span.     As  y  =  ii 

i 

,,         .,     .  /  \      •          -,-u      ^      75>°oo  105,000 

the  cntenon,  eq.    (5),  gives  either  7-=-^-     -  or,  -         — , 

I      367,500         367,500 

the  first  being  too  small  and  the  second  too  large.  Hence 
W±  at  the  quarter-point  is  the  proper  position  for  the 
maximum  bending  moment.  W\  will  be  84  ft.  from  the 
right-hand  end  of  the  span.  Taking  moments  of  all  the 
wheels  about  that  point,  by  the  aid  of  Table  I,  the  reac- 
tion R  at  the  left  end  of  the  span  is : 

,-,     14,830.000 

R  =  -  -  =  168,500  Ibs. 

88 


92  FLEXURE.  [Ch.  II. 

Eq.  (6)  will  then  give  the  bending  moment  at  Wi, 
but  having  the  reaction  R  and  using  Table  I  the  bending 
moment  becomes: 

M  =  168,500 X22  —720,000  =  2,987,000  ft.-lbs. 

The  end  shear  with  the  load  placed  so  as  to  produce 
the  greatest  bending  moment  at  the  quarter-point  is  ob- 
viously the  reaction  R  =  1 68, 500  Ibs.  The  shear  immediately 
at  the  left  of  the  quarter-point  will  be  168,500  —  75,000 
=  93,500  Ibs. 

The  greatest  bending  moment  at  the  centre  of  span 
is  similarly  found.  If  Wis  be  placed  at  the  centre  of  the 
span  the  wheel  weights  W&  .  .  .  Wis  and  9  ft.  in  length 
of  the  uniform  train  load  will  rest  on  the  span.  The  ratio 

representing  the  criterion,  eq.  (5),  is  y  =  — 5-  or  — r.     The 

I     310        310 

first  of  these  values  is  too  large  and  the  latter  is  too  small, 
showing  that  Wis  at  the  centre  of  the  span  is  the  correct 
position  for  the  greatest  bending  moment  at  that  point. 
The  reaction  R  for  this  position  of  the  load  is  at  once 
written  by  the  aid  of  Table  I  as  follows: 

11,695  +  2619  +  121.5 
R  =  —  -  X 1000  =  164,000  Ibs. 

oo 

The  bending  moment  M  for  the  centre  of  the  span  is  as 
follows,  using  the  preceding  value  of  R  and  Table  I : 

Af  =(1X14,440.5— 3370)  X 1000  =3, 848, ooo  ft.-lbs. 

The  end  shear  for  this  position  of  the  loading  is  the 
reaction  R,  i.e.,  164,000  Ibs.  The  shear  indefinitely  near 
to  but  at  the  left  of  the  centre  is  164,000  —  153,000  =  11,000 
Ibs,  This  small  shear  shows  that  the  moment  at  the  cen- 


Art.  21.]     BENDING  MOMENT  OF  A  NON-CONTINUOUS  BEAM.         93 

tre  of  the  span  is  the  greatest  in  the  entire  span  for  this 
position  of  loading. 

Assembling  the  preceding  results,  the  total  dead  and 
moving  load  moments  and  shears  will  be  as  follows: 

Moments. 

Quarter-point.  Centre. 

Dead  Load 654,000  ft.-lbs.  871,000  ft.-lbs. 

Moving  Load.  .  .  .2,987,000  ft.-lbs.  3,848,000  ft.-lbs. 


3,641,000  ft.-lbs.          4,719,000  ft.-lbs. 

Shears. 

End.  Quarter-point.  Centre. 

Dead  Load 39,600  Ibs.         19,800  Ibs.  zero 

Moving  Load. .  .  .  168,500  Ibs.         93,500  Ibs.       11,000  Ibs. 


Total 208,100  Ibs.       113, 300  Ibs.       n,ooolbs. 

The  expression  "  equivalent  uniform  load,"  for  moments 
or  shears,  as  the  case  may  be,  is  sometimes  used.  It 
simply  means  that  the  uniform  load  is  such  as  to  produce 
the  moments  or  shears  equivalent  to  those  found  under 
given  conditions.  A  uniform  load  p  per  linear  foot  acting 

on  the  entire  span  /  will  produce  a  centre-moment  of  — . 

8 

pi2 
Hence  if  there  be  written  -3- =  3, 848,000,  then,  if  /  =  88: 

o 
o 

p= X3,848, 000  =  3980  Ibs.  per  linear  foot. 

The  equivalent  uniform  load  therefore  for  the  greatest 
bending  moment  at  the  centre  of  the  span  is  3980  Ibs. 
per  linear  foot.  Similarly  as  the  bending  moment  at  any 


94  FLEXURE.  [Ch.  II. 

distance  x  from  one  end  of  the  span  is  ^(loc-oc2),  if  oo  be  made 

2 

22  in  the  present  case,  I  being  88  feet,  there  will  be  found 
by  placing  this  expression  equal  to  2,987,000  ft.-lbs: 


2,987,000  „  ...          ,. 

—  —     i— —  =4114  Ibs.  per  linear  foot. 
726 


The  end  shear  for  a  uniform  load  over  the  whole  span 
is  equal  to  the  load  on  half  the  span.     Hence  by  placing 
=  164,000  Ibs.,  there  will  result: 


164,000 

p=-        -=3727  Ibs.  per  linear  foot. 
44 


This  is  the  equivalent  uniform  load  for  the  end  shear 
with  the  load  so  placed  as  to  give  the  greatest  bending 
moment  at  the  centre  of  the  span. 

In  the  same  way  the  equivalent  uniform  load  for  the 
end  shear  168,600  Ibs.,  with  the  load  placed  so  as  to  give 
the  greatest  bending  moment  at  the  quarter-point,  will 
be  found  to  be  3830  Ibs.  per  linear  foot. 

These  simple  instances  show  that  the  equivalent  uni- 
form load  varies  from  one  case  to  another  according  to 
the  amount,  distribution  and  position  of  the  loading. 

Art.  22. — Moments  and  Shears  in  Special  Cases. 

Certain  special  cases  of  beams  are  of  such  common 
occurrence,  and  consequently  of  such  importance,  that  a 
somewhat  more  detailed  treatment  than  that  already 
given  may  be  deemed  desirable.  The  following  cases  are 
of  this  character: 


Art.  22.]        MOMENTS  AND  SHEARS  IN  SPECIAL  CASES.  95 

Case  I. 

Let  a  non -continuous   beam  supporting  a  single  weight 

P  at  any  point  be  con- 
sidered, and  let  such  a 
beam  be  represented  in 
Fig.  i.  If  the  span  RR' 
is  represented  by 

the  reactions  R  and  Rf  will  be 

R=jP,     and     R'=-jP.      .     .     .     .     (i) 

Consequently,  if  x  represents  the  distance  of  any  sec- 
tion in  RP  from  R,  while  xf  represents  the  distance  of  any 
section  of  R'P  from  R',  the  general  values  of  the  bending 
moments  for  the  two  segments  a  and  b  of  the  beam  will  be 

M  =  Rx,     and     M'=R'x'.    .....     (2) 

These  two  moments  become  equal  to  each  other  and 
represent  the  greatest  bending  moment  in  the  beam  when 

x=a     and     x'  =  6, 

or  when  the  section  is  taken  at  the  point  of  application  of-  the 
load  P. 

Eq.  (2)  shows  that  the  moments  vary  directly  as  the 
distances  from  the  ends  of  the  beam.  Hence  if  AP  (nor- 
mal to  RR')  is  taken  by  any  convenient  scale  to  represent 

the  greatest  moment,   -yP,   and  if   RARf  is  drawn,  any 

intercept  parallel  to  AP  and  lying  between  RAR'  and  RRf 
will  represent  the  bending  moment  for  the  section  at  its 
foot  by  the  same  scale.  In  this  manner  CD  is  the  bend- 
ing moment  at  D. 

The  shear  is  uniform  for  each  single   segment;    it  is 


96 


FLEXURE. 


[Ch.  II. 


evidently  equal  to  R  for  RP  and  R'  for  R'P.     It  becomes 
zero  at  P,  where  is  found  the  greatest  bending  moment. 

Case  II. 

Again,  let  Fig.  2  represent  the  same  beam  shown  in 
Fig.  i,  but  let  the  load  be  one  of  uniform  intensity,  p, 
extending  from  end  to  end  of  the  beam.  Let  C  be  placed 
at  the  centre  of  the  span, 
and  let  R  and  R' ',  as  before, 
represent  the  two  reactions. 
Since  the  load  is  symmetri- 
cal in  reference  to  C, 

R=R. 

For  the  same  reason  the 
moments  and  shears  in  one 
half  of  the  beam  will  be  exactly  like  those  in  the  other; 
consequently  reference  will  be  made  to  one  half  of  the 
beam  Only.  Let  oc  and  XL  then  be  measured  from  R 
toward  C.  The  forces  acting  upon  the  beam  are  R  and 
p,  the  latter  being  uniformly  continuous.  Applying  the 
formulae  for  the  bending  moment  at  any  section  x,  re- 
membering that  x1  has  all  values  less  than  x, 


FIG.  2. 


M=Rx-pf    (x-xjdi 
*/  o 


i» 


/.  M=Rx- 


If  /  is  the  span,  at  C,  M  becomes 


M          - 
•«      2        8 


(3) 


(4) 


But  because  the  load  is  unif6rm 


Art.  22.]        MOMENTS  AND  SHEARS  IN  SPECIAL  CASES.  97 

Hence 

M     PP-WI  M 

Mi  =  -£--^ W 


if  W  is  put  for  the  total  load.     Placing 

2  ' 
in  eq.  (3), 


(6) 


The  moments  M,  therefore,  are  proportional  to  the 
abscissae  of  a  parabola  whose  vertex  is  over  C,  and  which 
passes  through  the  origin  of  coordinates  R.  Let  AC,  then, 
normal  to  RR',  be  taken  equal  to  Mv  and  let  the  parabola 
RAR'  be  drawn.  Intercepts,  as  FH,  parallel  to  AC,  will 
represent  bending  moments  in  the  sections,  as  //,  at  their 
feet. 

The  shear  at  any  section  is 

S  =  —  =R-*x  =  4>(--x} 

dx          f?    P\2     /'  •    •    •    •    w 

or  it  is  equal  to  the  load  covering  that  portion  of  the  beam 
between  the  section  in  question  and  the  centre. 

Eq.  (7)  shows  that  the  shear  at  the  centre  is  zero;  it 
also  shows  that  S  =  R  at  the  ends  of  the  beam.  It  further 
demonstrates  that  the  shear  varies  directly  as  the  distance 
from  the  centre.  Hence,  take  RB  to  represent  R  and  draw 
EC.  The  shear  at  any  section,  as  Ht  will  then  be  repre- 
sented by  the  vertical  intercept,  as  HG,  included  between 
EC  and  RC. 

The  shear  being  zero  at  the  centre,  the  greatest  bending 
moment  will  also  be  found  at  that  point.  This  is  also 
evident  from  inspection  of  the  loading.  ' 

Eq.  (2)  of  Case  I  shows  that  if  a  beam  of  span  /  carries  a 


98 


FLEXURE. 


[Ch.  II. 


w 

weight  —  at  its  centre,  the  moment  M  at  the  same  point 
will  be 


(8) 


M  -—  1=  — 

1  ~  4  '  2       8  * 

The  third  member  of  eq.  (8)  is  identical  with  the  third 
member  of  eq.  (5).  It  is  shown,  therefore,  that  a  load 
concentrated  at  the  centre  of  a  non-continuous  beam  will 
cause  the  same  moment,  at  that  centre,  as  double  the  same 
load  uniformly  distributed  over  the  span. 

Eqs.  (5)  and  (8)  are  much  used  in  connection  with  the 
bending  of  ordinary  non-continuous  beams,  whether  solid 
or  flanged ;  and  such  beams  are  frequently  found. 

Case  HI. 

The  third  case   to    be  taken  is  a  cantilever  uniformly 
loaded;    it  is  shown  in  Fig.  3.     Let 
x  be  measured  from  the  free  end  A, 
and  let  the  uniform  intensity  of  the 
load  be  represented  by  p.     The  load  | 
px  acts  with  its  centre  at  the  distance 
%x  from  the  section  x.     Hence  the 
desired  moment  will  be 

M=—px.-= .    .     (9) 

If  AB  =  /,  the  moment  at  B  is 


FIG.  3. 


do) 


The  negative  sign  is  used  to  indicate  that  the  lower  side 
of  the  beam  is  subjected  to  compression.  In  the  two  pre- 
ceding cases,  evidently  the  upper  side  is  in  compression. 

The  shear  at  any  section  is 


Art.  23.]        FORMULA  OF  COMMON  THEORY  OF  FLEXURE.  99 

Hence  the  shear  at  any  section  is  the  load  between  the  free 
end  and  that  section. 

Eq.  (9)  shows  that  the  moments  vary  as  the  square 
of  the  distance  from  the  free  end;  consequently  the 
moment  curve  is  a  parabola  with  the  vertex  at  A,  and 
with  a  vertical  axis.  Let  EC,  then,  represent  M1  by  any 
convenient  scale  and  draw  the  parabola  CD  A.  Any  ver- 
tical intercept,  as  DF,  will  represent  the  moment  at  the 
section,  as  jp,  at  its  foot. 

Again,  let  EG  represent  the  shear  pi  at  B,  then  draw 
the  straight  line  AG.  Any  vertical  intercept,  as  HF,  will 
then  represent  the  shear  at  the  corresponding  section  F. 


Art.  23. — Recapitulation  of  the  General  Formulae  of  the 
Common  Theory  of  Flexure. 

It  is  convenient  for  many  purposes  to  arrange  the 
formulse  of  the  Common  Theory  of  Flexure  in  the  most 
general  and  concise  form.  In  this  article  the  preceding 
general  formulse  for  shear,  strains,  resisting  moments,  and 
deflections  will  be  recapitulated  and  so  arranged.  In 
order  to  complete  the  generalization,  the  summation  sign  2 
will  be  used  instead  of  the  sign  of  integration. 

In  Fig.  i,  let  ABC  represent  the  centre  line  of  any  bent 
beam;  AF,  a  vertical  line  through  A ;  CF,&  horizontal  line 
through  C,  while  A  is  the  section  of  the  beam  at  which  the 
deflection  (vertical  or  horizontal)  in  reference  to  C,  the 
bending  moment,  the  shearing  stress,  etc.,  are  to  be  deter- 
mined. As  shown  in  figure,  let  x  be  the  horizontal  coor- 
dinate measured  from  A,  and  y  the  vertical  one  measured 
from  the  same  point ;  then  let  xi  be  the  horizontal  distance 
from  the  same  point  to  the  point  of  application  of  any 
external  vertical  force  P.  To  complete  the  notation,  let  D 


ioo  FLEXURE.  [Ch.  II. 

be  the  deflection  desired;  Mi,  the  moment  of  the  external 
forces  about  A\  S,  the  shear  at  A;  u,  the  strain  (exten- 


"F~" 


FIG.  i. 

sion  or  compression)  per  unit  of  length  of  a  fibre  parallel  to 
the  neutral  surface  and  situated  at  a  normal  distance  of 
unity  from  it;  /,  the  general  expression  of  the  moment  of 
inertia  of  a  normal  cross-section  of  the  beam,  taken  in 
reference  to  the  neutral  axis  of  that  section  ;  E,  the  coeffi- 
cient of  elasticity  for  the  material  of  the  beam  ;  and  M  the 
moment  of  the  external  forces  for  any  section,  as  B. 

Again,  let  A  be  an  indefinitely  small  portion  of  any 
normal  cross-section  of  the  beam,  and  let  z  be  an  ordinate 
normal  to  the  neutral  axis  of  the  same  section.  By  the 
11  common  theory  "  of  flexure,  the  intensity  of  stress  at  the 
distance  z  from  the  neutral  surface  is  (zP'E).  Conse- 
quently the  stress  developed  in  the  portion  'A  of  the  sec- 
tion is  EP'zA,  and  the  resisting  moment  of  that  stress 
is  EP'z^. 

The  resisting  moment  of  the  whole  section  will  there- 
fore be  found  by  taking  the  sum  of  all  such  moments  for 
its  whole  area. 

Hence 


Hence,  also, 

-       M 
M==EI' 


Art.  23.]          FORMULA  OF  COMMON  THEORY  OF  FLEXURE.          101 

If  n  represents  an  indefinitely  short  portion  of  the 
neutral  surface,  the  strain  for  such  a  length  of  fibre  at  unit's 
distance  from  that  surface  will  be  nu. 

If  the  beam  were  originally  straight  and  horizontal,  n 
would  be  equal  to  dx. 

u  being  supposed  small,  the  effect  of  the  strain  nu  at 
any  section,  B,  is  to  cause  the  end  A  of  the  chord  B  A  to 
move  vertically  through  the  distance  nux. 

If  BK  and  BA  (taken  equal)  are  the  positions  of  the 
chords  before  and  after  flexure,  nux  will  be  the  vertical 
distance  between  K  and  A. 

By  precisely  the  same  kinematical  principle  the  ex- 
pression nuy  will  be  the  horizontal  movement  of  A  in 
reference  to  B. 

Let  Inux  and  Inuy  represent  summations  extending 
from  A  to  C,  then  will  those  expressions  be  the  vertical  and 
horizontal  deflections  respectively  of  A  in  reference  to  C. 
It  is  evident  that  these  operations  are  perfectly  general, 
and  that  x  and-  y  may  be  taken  in  any  direction  whatever. 

The  following  general  but  strictly  approximate  equa- 
tions relating  to  the  subject  of  flexure  may  now  be  written : 

S=ZP d) 

--2Pxi.  .....     (2) 

M 


— (4) 


f  ^ 
(5) 


102  FLEXURE.  [Ch.  II. 


Dh  represents  horizontal  deflection. 

The  summation  2Pz  must  extend  from  A  to  a  point  of 
no  bending,  or  from  A  to  a  point  at  which  the  bending 
moment  is  M/.  In  the  latter  case 


.     .     (7) 
Mi'  may  be  positive  or  negative. 

Art.  24.  —  The  Theorem  of  Three  Moments. 

The  object  of  this  theorem  is  the  determination  of  the 
relation  existing  between  the  bending  moments  which  are 
found  in  any  continuous  beam  at  any  three  adjacent  points 
of  support.  In  the  most  general  case  to  which  the  theorem 
applies,  the  section  of  the  beam  is  supposed  to  be  variable, 
the  points  of  support  are  not  supposed  to  be  in  the  same 
level,  and  at  any  point,  or  all  points,  of  support  there  may 
be  constraint  applied  to  the  beam  external  to  the  load 
which  it  is  to  carry  ;  or,  what  is  equivalent  to  the  last  con- 
dition, the  beam  may  not  be  straight  at  any  point  of  sup- 
port before  flexure  takes  place. 

Before  establishing  the  theorem  itself,  some  prelimi- 
nary matters  must  receive  attention. 

If  a  beam  is  simply  supported  at  each  end,  the  reactions 
are  found  by  dividing  the  applied  loads  according  to  the 
simple  principle  of  the  lever.  If,  however,  either  or  both 
ends  are  not  simply  supported,  the  reaction  in  general  is 
greater  at  one  end  and  less  at  the  other  than  would  be 
found  by  the  law  of  the  lever;  a  portion  of  the  reaction  at 
one  end  is,  as  it  were,  transferred  to  the  other.  The  trans- 


Art.  24.] 


THE  THEOREM  OF  THREE  MOMENTS. 


103 


ference  can  only  be  accomplished  by  the  application  of  a 
couple  to  the  beam,  the  forces  of  the  couple  being  applied 
at  the  two  adjacent  points  of  support;  the  span,  conse- 
quently, will  be  the  lever-arm  of  the  couple.  The  existence 
of  equilibrium  requires  the  application  to  the  beam  of  an 
equal  and  opposite  couple.  It  is  only  necessary,  however, 
to  consider,  in  connection  with  the  span  AB,  the  one  shown 
in  Fig.  i.  Further,  from  what  has  immediately  preceded, 


FIG.  i. 

it  appears  that  the  force  of  this  couple  is  equal  to  the 
difference  between  the  actual  reaction  at  either  point  of 
support  and  that  found  by  the  law  of  the  lever.  The 
bending  caused  by  this  couple  may  evidently  be  of  an 
opposite  kind  to  that  existing  in  a  beam  simply  supported 
at  each  end. 

These  results  are  represented  graphically  in  Fig.  i.  A 
and  B  are  points  oi  support,  and  AB  is  the  beam;  AR  and 
BRf  are  the  reactions  according  to  the  law  of  the  lever; 
RF  =  R'F  is  the  force  of  the  applied  couple ;  consequently 

AF=AR  +  RF    and     BF  =  BRf  -  (R'F  =  RF) 

are  the  reactions  after  the  couple  is  applied.  As  is  well 
known,  lines  parallel  to  CK,  drawn  in  the  triangle  ACB, 


104 


FLEXURE. 


[Ch.  II. 


represent  the  bending  moments  at  the  various  sections  of 
the  beam,  when  the  reactions  are  AR  and  BR' '.  Finally, 
vertical  lines  parallel  to  AG,  in  the  triangle  QHG,  will 
represent  the  bending  moments  caused  by  the  force  R'F. 

In  tfre  general  case  there  may  also  be  applied  to  the 
beam  two  equal  and  opposite  couples  having  axes  passing 
through  A  and  B  respectively.  The  effect  of  such  couples 
will  be  nothing  so  far  as  the  reactions  are  concerned,  but 
they  will  cause  uniform  bending  between  A  and  B.  This 


FIG.  2, 


FIG.  3. 

•uniform  or  constant  moment  may  be  represented  by  ver- 
tical lines  drawn  parallel  to  AH  or  LN  (equal  to  each 
other)  between  the  lines  A  B  and  HQ.  The  resultant 
moments  to  which  the  various  sections  of  the  beam  are 
subjected  will  then  be  represented  by  the  algebraic  sum 
of  the  three  vertical  ordinates  included  between  the  lines 
ACB  and  GQ.  Let  that  resultant  be  called  M.  This 
composition  of  the  resultant  moment  M  will  be  made 
clearer  by  reference  to  Figs.  2  and  3.  Fig.  2  shows  the 
component  moment  due  to  the  single  force  F  acting  with 


Art.  24.]  THE   THEOREM  OF  THREE  MOMENTS.  105 

the  lever-arm  /  so  that  its  moment  increases  directly  as 
the  distance  from  B.  Fig.  3,  on  the  other  hand,  shows  the 
component  moment  due  to  the  two  equal  and  opposite 
couples  acting  at  the  ends  of  the  span.  The  resultant 
moment  M  is  the  algebraic  sum  of  the  three  component 
moments,  shown  combined  in  Fig.  i. 

Let  the  moment  GA  be  called  Ma,  and  the  moment 

BQ=LN=HA=Mb. 

Also  designate  the  moment  caused  by  the  load  P,  shown 
by  lines  parallel  to  CK  in  ACB,  by  Mr  Then  let  x  be  any 
horizontal  distance  measured  from  A  toward  B\  I  the 
horizontal  distance  A  B  ;  and  z  the  distance  of  the  point  of 
application,  K,  of  the  force  P  from  A.  With  this  nota- 
tion there  can  be  at  once  written 


.  (i) 


Eq.  (i)  is  simply  the  general  form  of  eq.  (2),  Art.  23. 

It  is  to  be  noticed  that  Fig.  i  does  not  show  all  the 
moments  M«,  M&,  and  Ml  to  be  the  same  sign,  but  for 
convenience  they  are  so  written  in  eq.  (i). 

The  formula  which  represents  the  theorem  of  three 
moments  can  now  be  written  without  difficulty.  The 
method  to  be  followed  involves  the  improvements  added 
by  Prof.  H.  T.  Eddy,  and  is  the  same  as  that  given  by  him 
in  the  "American  Journal  of  Mathematics,"  Vol.  I.,  No.  i. 

Fig.  4  shows  a  portion  of  a  continuous  beam,  including 
two  spans  and  three  points  of  supports.  The  deflections 
will  be  supposed  'measured  from  the  horizontal  line  NQ. 
The  spans  are  represented  by  la  and  lc\  the  vertical  dis- 

*  This  equation  is  used  in  the  next  Art.  for  a  short  demonstration  of 
the  common  form  of  the  Theorem  of  Three  Moments. 


io6 


FLEXURE. 


[Ch.  II. 


tances  of  NQ  from  the  points  of  support  by  ca,  cb,  and  cc\ 
the  moments  at  the  same  points  by  M«,  Mb,  and  Mc,  while 
the  letters  5  and  R  represent  shears  and  reactions  re- 
spectively. 

In  order  to  make  the  case  general,  it  will  be  supposed 
that  the  beam  is  curved  in  a  vertical  plane,  and  has  an 


FIG.  4. 

elbow  at  b,  before  flexure,  and  that,  at  that  point  of  sup- 
port, the  tangent  of  its  inclination  to  a  horizontal  line, 
toward  the  span  la,  is  t,  while  t'  represents  the  tangent  on 
the  other  side  of  the  same  point  of  support  ;  also  let  d  and 
d!  be  the  vertical  distances,  before  bending  takes  place,  of 
the  points  a  and  c,  respectively,  below  the  tangents  at  the 
point  b. 

A  portion  of  the  difference  between  ca  and  cb  is  due  to 
the  original  inclination,  whose  tangent  is  t,  and  the  original 
lack  of  straightness,  and  is  not  caused  by  the  bending; 
that  portion  which  is  due  to  the  bending,  however,  is, 
remembering  eq.  (5),  Art.  23, 

Mxn 


Fig.  5  will  make  clear  the  component  parts  of  the  value 
of  D  in  the  preceding  equation. 

By  the  aid  of  eq.  (i)  this  equation  may  be  written: 
E(ca-cb-lat-d) 


Art.  24.] 


THE  THEOREM  OF  THREE  MOMENTS. 


107 


In  this  equation,  it  is  to  be  remembered,  both  x  and  z 
(involved  in  Mx)   are  measured  from   support  a  toward 


FIG.  5. 

support  b.     Now  let  a  similar  equation  be  written  for  the 
span  lc>  in  which  the  variables  x  and  z  will  be  measured 
from  c  toward  b.     There  will  then  result 
E(cc-cb-lct'-d') 


When  the  general  sign  of  summation  is  displaced  by 
the  integral  sign,  n  becomes  the  differential  of  the  axis  of 
the  beam,  or  ds.  But  ds  may  be  represented  by  udx,  u 
being  such  a  function  of  x  as  becomes  unity  if  the  axis  of 
the  beam  is  originally  straight  and  parallel  to  the  axis  of  x. 
The  eqs.  (2)  and  (3)  may  then  be  reduced  to  simpler  forms 
by  the  following  methods:* 

*  These  analytic  transformations  are  of  the  nature  of  convenient  but 
arbitrary  notation  and  are  not  to  any  degree  whatever  analytic  demon- 
strations. 


io8  FLEXURE.  [Ch.  II. 

In  eq.  (2)  put 

afl-x\xn     £    fau(la-x}xdx     oc 

2~~      '  - 

Also 


Xa    fau(la-x)dx        iaXa    f 

rjb    ~r  —  IT./* 


Also 


In  the  same  manner 

x2dx    %d  Ca  uxd% 


Also 


And 


xaf   r«uxdx     ia'xa' 


y.  r   u  r    .. 

"a  -^a      I  1           "a  -"a  ^a      I  j              *a  -*a  ^a  "a                /    v 

-;  —  /  ^txdx=  -  -,  -  •  /  xdx=  -  .    .     (o) 

l        J*  l           J*>  2 


a  a  2 

Again,  in  the  same  manner, 

al\/T 
— 


aU^M.xJx  .....       (10) 
b       1 

Using  eqs.  (4)  to  (10),  eq.  (2)  may  be  written: 


iatiallMlxJx.      (n) 


Art.  24.]  THE  THEOREM  OF  THREE  MOMENTS.  109 

Proceeding  in  precisely  the  same  manner  with  the  span 
lc>  ecl-  (3)  becomes 


leSM\x,Ax.      (12) 

b 

The  quantities  xa  and  xc  are  to  be  determined  by  apply- 
ing eq.  (4)  to  the  span  indicated  by  the  subscript;  while 
ua,  ia>  uc,  and  if  are  to  be  determined  by  using  eqs.  (5)  and 
(6)  in  the  same  way.  Similar  observations  apply  to  iia't 
id  •>  Xa,  u'  ,  ij,  and  ocj  taken  in  connection  with  eqs.  (7), 
(8),  and  (9). 

If  /  is  not  a  continuous  function  of  x,  the  various  inte- 
grations of  eqs.  (4),  (5),  (7),  and  (8)  must  give  place  to 
summation:  (I)  taken  between  the  proper  limits. 

Dividing  eqs.  (n)  and  (12)  by  la  and  lc  respectively, 
and  adding  the  results, 


ce-cb  d     df\ 

~T~  ~    Ta~~r) 


M 


+  J  (Mauaiaxa 


(13) 


in  which  T  = 

Eq.  (13)  is  the  most  general  form  of  the  theorem  of 
three  moments  if  E,  the  coefficient  of  elasticity,  is  a  con- 
stant quantity.  Indeed,  that  equation  expresses,  as  it 
stands,  the  "  theorem  "  for  a  variabL  coefficient  of  elas- 
ticity if  (ie)  be  written  instead  of  i\  e  representing  a  quan- 
tity determined  in  a  manner  exactly  similar  to  that  used 
in  connection  with  the  quantity  i. 


no  FLEXURE.  [Ch.  II. 

In  the  ordinary  case  of  an  engineer's  experience  r=o, 
d  =  df=o,  I  =  constant,  u=ua=uc=etc.,=c' =  secant  of  the 
inclination  for  which  t  =  —  t'  is  the  tangent;  consequently 


*    = 


From  eq.  (4) 


From  eq.  (7) 


6  ' 


The  summation  2MjcAx  can  be  readily  made  by  refer- 
ring to  Fig.  i. 

The  moment  represented  by  CK  in  that  figure  is 


consequently  the  moment  at  any  point  between  A  and  K, 
due  to  P,  is 


. 

I 

Between  K  and  B 


-z\  x 
7-)  .z.- 
I  J  z 


Using  these  quantities  for  the  span  /a, 
2MjcAx=    I   M1xdx+    /  aMt'xdx  = 

b  J  0  «/2 


Art.  24.]  THE  THEOREM  OF    THREE   MOMENTS.  in 

For  the  span  lc  the  subscript  a  is  to  be  changed  to  c. 
Introducing  all  these  quantities  eq.  (13)  becomes,  aftei 
providing  for  any  number  of  weights,  P: 


Eq.  (14),  with  c'  equal  to  unity,  is  the  form  in  which  the 
theorem  of  three  moments  is  usually  given;  with  c'  equal 
to  unity  or  not,  it  applies  only  to  a  beam  which  is  straight 
before  flexure,  since 


If  such  a  beam  rests  on  the  supports  a,  b,  and  c,  before 
bending  takes  place, 


and  the  first  member  of  eq.  (14)  becomes  zero. 

If,  in  the  general  case  to  which  eq,  (13)  applies,  the 
deflections  ca,  cb,  and  cc  belong  to  the  beam  in  a  position 
of  no  bending,  the  first  member  of  that  equation  disappears, 
since  it  is  the  sum  of  the  deflections  due  to  bending  only 
for  the  spans  la  and  lc,  divided  by  those  spans,  and  each 
of  those  quantities  is  zero  by  the  equation  immediately 
preceding,  eq.  (2).  Also,  if  the  beam  or  truss  belonging 
to  each  span  is  straight  between  the  points  of  support 
(such  points  being  supposed  in  the  same  level  or  not)  ,  ua  = 
'Haf  =  ula-=  constant,  and  uc=urf  =  ulc=  another  constant.  If, 
finally,  7  be  again  taken  as  constant,  oca  and  xc,  as  well  as 
Ax,  will  have  the  values  found  above. 

From  these  considerations  it  at  once  follows  that  the 


H2  FLEXURE.  [Ch.II. 

second  member  of  eq.  (14),  put  equal  to  zero,  expresses 
the  theorem  of  three  moments  for  a  beam  or  truss  straight 
between  points  of  support,  when  those  points  are  not  in 
the  same  level,  but  when  they  belong  to  a  configuration 
of  no  bending  in  the  beam.  Such  an  equation,  however, 
does  not  belong  to  a  beam  not.  straight  between  points  of 
support. 

The  shear  at  either  end  of  any  span,  as  /a,  is  next 
to  be  found,  and  it  can  be  at  once  written  by  referring  to 
the  observations  made  in  connection  with  Fig.  i.  It  was 
there  seen  that  the  reaction  found  by  the  simple  law  o.f 
the  lever  is  to  be  increased  or  decreased  for  the  continuous 
beam,  by  an  amount  found  by  dividing  the  difference  of 
the  moments  at  the  extremities  of  any  span  by  the  span 
itself.  Referring,  therefore,  to  Fig.  4,  for  the  shears  5, 
there  may  at  once  be  written:* 


The  negative  sign  is  put  before  the  fraction 


M  -Mt 


in  eq.  (15)  because  in  Fig.  i  the  moments  Ma  and  Mb  are 
represented  opposite  in  sign  to  that  caused  by  P,  while  in 


Art.  24.]  THE  THEOREM    OF    THREE   MOMENTS.  113 

eq.  (i)  the  three  moments  are  given  the  same  sign,  as  has 
already  been  noticed. 

Eqs.  (15)  to  (18)  are  so  written  as  to  make  an  upward 
reaction  positive,  and  they  may,  perhaps,  be  more  simply 
found  by  taking  moments  about  either  end  of  a  span.  For 
example,  taking  moments  about  the  right  end  of  la, 


From  this,  eq.  (15)  at  once  results.      Again,  moments 
about  the  left  end  of  the  same  span  give 


This  equation  gives  eq.  (16),  and  the  same  process  will 
give  the  others. 

If  the  loading  over  the  different  spans  is  of  uniform 
intensity,  then,  in  general,  P  =  wdz,  w  being  the  intensity. 
Consequently 

/;  Z4 

w(l2-z2)zdz=w—  . 
4 

In  all  equations,   therefore,   for 


I   3 

chere  is  to  be  placed  the  term  «;a—  ;  and  for 

4 


1 5 

the  term  wc— .     The  letters  a  and  c  mean,  of  course,  that 

4 

reference  is  made  to  the  spans  la  and  lc. 


H4  FLEXURE.  [Ch.  II. 

From  Fig.  4,  there  may  at  once  be  written: 

•R   =SB'+5a .     (19) 

R'  =S0'+S&.     ......     (20) 

R"=S,'+Sc,    .     .     .   \     .    '.     (21) 
etc.  =etc.  +etc. 

Art.  25. — Short  Demonstration  of  the  Common  Form  of  the 
Theorem  of  Three  Moments. 

The  general  demonstration  of  the  Theorem  of  Three 
Moments  given  in  the  preceding  article  has  the  great 
advantage  of  showing  the  influence  of  all  the  elements 
which  enter  the  complete  problem,  including  variability  of 
moment  of  inertia,  lack  of  straightness  of  beam,  and  points 
of  support  not  at  the  same  elevation.  An  adequate  con- 
ception of  the  influences  of  the  assumptions  made  in  estab- 
lishing the  common  or  approximate  form  of  the  theorem 
can  be  obtained  only  by  the  employment  of  the  general 
analysis,  but  it  is  convenient  to  establish  the  usual  or 
approximate  form  of  the  theorem  by  a  short  direct  method 
like  the  following. 

Eq.  (i)  of  the  preceding  article  gives  the  general  value 
of  the  bending  moment  in  any  span  whatever  of  a  con- 
tinuous beam  such  as  that  shown  in  Fig.  i.  The  notation 
given  in  that  figure  explains  itself  and  is  essentially  the  same 
as  that  already  used.  It  should  be  remembered  that  each 
reaction  R,  R',  and  R"  is  composed  of  two  shears  as  indi- 
cated, one  acting  at  an  indefinitely  short  distance  to  the 
left  of  a  point  of  support  and  the  other  at  an  indefinitely 
short  distance  to  the  right  of  the  same  support.  It  is 
supposed  that  one  load  acts  in  each  span  at  the  distance 


Art.  25.]    COMMON  FORM  OF  THEOREM  OF  THREE  MOMENTS.      115 

z  from  the   left-hand  end  of  the  left-hand   span,  or  from 
the  right-hand  end  of  the  right-hand  span. 

Using  eq.  (i)  of  the  preceding  article  and  representing 
the  deflection  at  any  point  in  the  span  l\  by  w,  eq.  (i)  may 
be  at  once  written: 


The  quantity  /i  is  the  moment  of  inertia  of  the  cross- 
section  of  the  beam  about  its  neutral  axis  and  E  is  the 


j.--*.^ 

1" 

P2 

—  1 

c  ! 

MO 

"1C  

m 


"^     '    .   "       "li  ~  ~^R'  \u 

FIG.  i. 


modulus  of  elasticity.  It  is  assumed  that  the  beam  is 
straight  and  horizontal  and  that  the  moment  of  inertia 
does  not  vary  in  either  span.  If  t\  is  the  tangent  of  the 
inclination  of  the  neutral  surface  of  the  beam  at  the  right- 
hand  end  of  the  span  /i,  then  integrating  eq.  (i)  between 
the  limits  of  x  and  /i  eq.  (2)  will  at  once  result: 

dw       i 


dx        fiJltfeV  2          2/        2/1^ 

The  integration   of  Midx  is  indicated  only  in  eq.   (2) 

for  the  reason  that  in  general  Mi  is  a  discontinuous  func- 

rh  rx 

tion.     The   double   integral   I     I  Mid#2   cannot   therefore 

Jo  Jh 

generally  be  completed  by  the  usual  procedures,   but  it 


n6  FLEXURE.  [Ch.  II. 


must  be  taken  as  ---InMioc,  as  given  by  eq.  (5)  of  Art. 
hi  i 

23.  The  value  of  this  expression  for  a  single  load  PI  is 
shown  in  detail  on  the  lower  half  of  page  1  10  of  the  preceding 
Art.  as  |Pi(/i2-,s2)£,  which  appears  in  eq.  (3).  By  integra- 
ting eq.  (2)  between  the  limits  of  l\  and  o,  remembering 
that  the  points  of  support  are  supposed  to  be  at  the  same 
elevation  and  hence  that  w  =  o  for  %  =  l\  : 


W=  —  -=r^(Mall+2Mi>li  H— T—  (h2  —  Z2)z]  +6/1  =O.         (3) 

Eli\  LI  i 

An  equation  identical  with  eq.  (3)  may  be  written  for 
the  right-hand  span  Z2  by  simply  changing  the  subscripts, 
remembering,  however,  that  the  origin  from  which  -z  and  x 
are  measured  is  the  point  of  support  C,  Fig.  i ,  and  that  the 
tangent  of  the  inclination  of  the  neutral  surface  at  the 
left-hand  end  of  the  span  h  will  be  —  ti. 

Hence : 


If  eqs.  (3)  and  (4)  be  added  the  usual  and  approximate 
form  of  the  Theorem  of  Three  Moments  will  at  once  result, 
except  that  the  moments  of  inertia  I\  and  1  2  are  different. 
Assuming  I\  =1%  and  writing  the  summation  sign  before 
PI  and  P2  to  indicate  that  any  number  of  loads  may  act 
on  every  span,  the  Theorem  of  Three  Moments  as  usually 
employed  will  at  once  result  : 


SP2(Pz-s?)z  ....     (5) 

12 


Art.  25.]     COMMON  FORM  OF  THEOREM  OF  THREE  MOMENTS.      117 

It  will  be  observed  that  eq.  (5)  is  identical  with  the 
second  member  of  eq.  (14)  of  the  preceding  article,  and  it 
is  the  equation  sought.  The  expressions  for  the  shears  com- 
posing each  of  the  reactions  may  now  easily  be  written. 

Taking  moments  about  the  right-hand  end  of  the  span  h  : 

Sali-2P1(ll-z)+Ma=M».   .     .     .     .     (6) 
Hence  : 


LI 


.     . 
(7) 


Again  taking  moments  about  the  left-hand  end  of  the 
same  span: 

S'J,i-2Piz+M*=Ma.       .     .].     .     (8) 
Hence  : 

9'      y-p  z   i  Ma-  Mi  (  . 

3l,  =  2^l--\  --  -  -  .  .       .        .       .        (9) 

LI  ll 

Eqs.  (7)  and  (g)  give  the  shears  at  the  two  ends  of  the 
span  /i  and  they  also  give  the  shears  at  the  two  ends  of 
the  span  1%  by  simply  changing  the  notation  so  as  to  apply 
to  the  span  1%  as  shown  in  eqs.  (10)  and  (n)  : 

z   .  Mc-Mb  ,     , 

--\  --  -  -  .....       (lO) 
/2  /2 


/2 


Each  reaction  will  be  the  sum  of  the  appropriate  pair 
of  shears  as  shown  by  eqs.  (19),  (20),  and  (21)  of  the  pre- 
ceding article. 

These  equations  are  given  in  their  most  general  forms; 


u8  FLEXURE.  [Ch.  II. 

that  is,  for  any  disposition  of  loads  of  any  magnitude.  They 
may  be  adapted  to  uniform  loading  either  partial  or  entire, 
as  indicated  on  the  lower  half  of  page  113. 


Art.  26.  —  Reaction  under  Continuous  Beam  of  any  Number 

of  Spans. 

The  general  value  of  the  reactions  at  the  points  of 
support  under  any  continuous  beam  have  been  given  in 
eqs.  (19),  (20),  (21),  etc.,  of  article  24.  Before  those 
equations,  however,  can  be  applied  to  any  particular  case, 
the  values  of  the  bending  moments,  which  appear  in  the 
expressions  50,  S&',  S&,  etc.,  for  the  shears,  must  be  deter- 
mined. In  the  application  of  the  theorem  of  three  mo- 
ments, it  is  usually  assumed  that  the  continuous  beam 
before  flexure  is  straight  between  the  points  of  support, 
and  that  the  latter  belong  to  a  configuration  of  no  bending. 
The  moment  of  inertia  I  is  also  assumed  to  be  constant. 
This  is  frequently  not  strictly  true,  yet  it  will  be  assumed 
in  what  follows,  since  the  method  to  be  used  in  finding 
the  moments  is  independent  of  the  assumption,  and  remains 
precisely  the  same  whatever  form  for  the  theorem  of  three 
moments  may  be  chosen. 

Agreeably  to  the  assumption  made,  eq.  (5)*  of  the  pre- 
ceding article  takes  the  following  form  : 


+  lc)  +Mclc  =  - 


*  Or  eq.  (14)  of  Art.  24. 


Art.  26.]      REACTIONS   UNDER  ANY  CONTINUOUS   BEAM.  119 

Let  Fig.   i   represent  a  continuous   beam   of  n  spans 
equal  or  unequal  in    length.     At   the    points   of    support, 


FIG.  i. 

o,  i,  2,  3,  4,  5,  etc.,  let  the  bending  moments  be  represented 
by  M0,  Mv  Mv  My  etc.  The  moment  M0  is  always  known ; 
it  is  ordinarily  zero,  and  that  will". be  considered  its  value. 

An  examination  of  Fig.  i  shows  that,  by  repeated 
applications  of  eq.  (i),  the  number  of  resulting  equations 
of  condition  will  be  one  less  than  the  number  of  spans. 
If  the  two  end  moments  are  known  (here  assumed  to  be 
zero),  the  number  of  unknown  moments  will  also  be  one 
less  than  the  number  of  spans.  Hence  the  number  of 
equations  will  always  be  sufficient  for  the  determination 
of  the  unknown  moments. 

For  the  sake  of  brevity  let  the  following  notation  be 
adopted : 


*1  "2 


etc.  =  etc.  -  etc. 


d3=l4. 
/4=/5. 


120  FLEXURE.  ]Ch.  II. 

i  denoting  any  number  of  the  series  i,  2,  3,  4,    ,  .  .  n.     It  is 
thus  seen  that,  in  general, 


also  that  a2=61,  c2  =  b3,  d3  =  c4,  etc.     These  relations  can  be 
used  to  simplify  the  final  result. 

By  repeated  applications  of  eq.  (i)  the  following  n 
equations  of  condition,  involving  the  notation  given  above, 
will  result: 


a2Mi  +b2M2+c2M3 


+/5M 


These  simultaneous  equations  may  be  treated  in  various 
ways  in  order  to  determine  the  values  of  the  moments  Mi, 
M2,  Ms,  etc.  The  preceding  notation  is  adapted  to  the 
method  by  determinants,  which  is  probably  as  simple  as 
any.  As  these  procedures  are  purely  algebraic  they  will 
not  be  further  developed  here. 

In  American  engineering  practice,  as  exemplified  in  the 
theory  of  revolving-swing  bridges,  it  is  necessary  to  con- 
sider at  most,  two  simultaneous  equations  of  condition 
whose  solution  requires  the  simplest  process  of  elimination 
only. 


Art.  27.]  DEFLECTION  BY  THE  COMMON  THEORY.  121 

This  last  case  may  be  simply  illustrated  by  referring 
to  Fig.  i,  in  which  M0  =o.  If  there  are  three  spans  Ma  =o 
as  one  of  the  end  spans.  The  first  two  of  eq.  (2)  will  be 
needed  : 

ui  ......     (3) 


U2  ......        (4) 

Simple  elimination  will  then  give: 


,.  ,,  ,  . 

Mi=  —  r  -  r;     and     M^  =  —  r  -  r~.       .     (s) 

0,102—0,201  a\02 


Reactions. 

After  the  moments  are  found,  either  by  the  general  or 
special  method,  for  any  condition  of  loading,  the  reactions 
will  at  once  result  from  the  substitution  of  the  values  thus 
found  in  the  eqs.  (15)  to  (21)  of  Art.  24,  which  it  is  not  neces- 
sary to  reproduce  here. 

Art.  27.  —  Deflection  by  the  Common  Theory  of  Flexure. 

The  deflection  or  sag  of  a  beam  subjected  to  loading  at 
right  angles  to  its  axis  is  the  displacement  of  the  neutral 
surface  in  the  direction  of  the  loading.  Ordinarily  the 
beam  is  horizontal  and  the  loading  vertical,  so  that  the 
deflection  is  also  vertical.  The  entire  deflection  is  due  both 
to  the  lengthening  and  the  shortening  of  the  fibres  on  the 
two  sides  of  the  netural  surface  and  to  the  action  of  the 
transverse  shear  throughout  the  beam.  The  equation 
leading  directly  to  the  former  portion  is  eq.  (7)  of  Art  14, 
but  the  equations  of  Art.  24  must  be  used  to  determine  the 
deflection  due  to  shear. 

Let  XQ  be  the  coordinate  of  some  point  at  which  the 


t**  FLBXURE.  [Ch.  II. 

tangent  of  the  inclination  of  the  neutral  surface  to  the  axis 
of  x  is  known;  then  from  eq.  (7)  of  Art.  14 


(i) 


-T-  will  be  at  once  recognized  as  the  general  value  of  the 

tangent  of  the  inclination  just  mentioned,  or,  in  the  case 
of  curved  beams,  as  approximately  the  difference  between 
the  tangent,  before  and  after  flexure. 

Again,  let  xl  represent  the  coordinate  of  a  point  at  which 
the  deflection  w  is  known,  then  from  eq.  (i)  : 


The  points  of  greatest  or  least  deflection  and  greatest 
or  least  inclination  of  neutral  surface  are  easily  found  by 
the  aid  of  eqs.  (i)  and  (2). 

The  point  of  greatest  or  least  deflection  is  evidently 
found  by  putting 

dw 


and  solving  for  x.     Since  -y-  is  the  value  of  the  tangent  of 

the  inclination  of  the  neutral  surface,  it  follows  that  a 
point  of  greatest  or  least  deflection  is  found  where  the  beam 
is  horizontal. 

Again,  the  point  at  which  the  inclination  will  be  great- 
est or  least  is  found  by  the  equation 


, 
adx 


Art.  27.]          DEFLECTION  BY   THE  COMMON   THEORY.  123 

But,  approximately,  -1-7  is  the  reciprocal  of  the  radius 

of  curvature  ;  hence  the  greatest  inclination  will  be  found 
at  that  point  at  which  the  radius  of  curvature  becomes  infi- 
nitely great,  or,  at  that  point  at  which  the  curvature  changes 
from  positive  to  negative  or  vice  versa.  These  points  are 
called  points  of  "contra  -flexure."  Since: 


there  is  no  bending  at  a  point  of  contra-fteocure. 

The  moment  of  the  external  forces,  M,  will  always  be 
expressed  in  terms  of  x.  After  the  insertion  of  such  values, 
eqs.  (i)  and  (2)  may  at  once  be  integrated  and  (3)  and  (4) 
solved. 

The  coefficient  of  elasticity,  E,  is  always  considered  a 
constant  quantity  ;  hence  it  may  always  be  taken  outside  the 
integral  signs.  In  all  ordinary  cases,  also,  /  is  constant 
throughout  the  entire  beam.  In  such-  cases,  then,  there 
will  only  need  to  be  integrated  the  expressions: 

/    Mdx     and      f*  f*Md&. 

J  x*  y.ft  Jx0 

It  is  sometimes  convenient  to  express  the  tangent  of 
inclination  of  the  neutral  surface  and  the  deflection  in 
terms  of  some  known  intensity  &0  of  fibre  stress  at  the 
distance  d  from  the  neutral  surface  and  at  a  section  of  the 
beam  where  the  known  external  bending  moment  is  M0. 
The  desired  expressions  may  readily  be  written  by  simply 
transforming  eqs.  (i)  and  (2)  to  the  proper  shape.  It 

has  been  shown  by  eq.  (10)  of  Art.  14  that  &0=-y-,  and 


124  FLEXURE.  [Ch.  II. 

hence  that  1  =  —^-.     By  substitution  of  this  value  of  / 

^0 

first  in  eq.  (i)  and  then  in  eq.  (2),  there  will  result: 


dw 


dx     EM0d/  Xo 
and 


/'*7,/r  ,  t  x 

/    Mtffc      .....     (5) 

JX 


(6) 


Eqs.  (5)  and  (6)  give  the  desired  expressions  in  which 
I  and  d  are  considered  constant  in  accordance  with  all 
ordinary  practice.  In  the  use  of  these  last  two  equations 
it  is  supposed  that  the  conditions  of  any  given  problems 
will  enable  kQ  and  M0  to  be  computed  as  known  quantities. 

The  general  form  of  the  integral  in  the  second  member 
of  eq.  (6)  is  easily  determined.  The  quantities  M0  and 
M  are  exactly  similar  expressions  with  the  same  number 
of  terms  and  of  the  same  degree.  The  effect  of  the  inte- 
gration of  M  twice  between  the  limits  indicated  is  to  raise 
the  degree  of  each  term  of  which  it  is  composed  by  two, 
so  that  the  double  integration  of  Mdx2  divided  by  M0  will 
be  a  simple  product  a/2,  a  being  a  numerical  quantity 
depending  upon  the  manner  of  loading,  the  condition  of 
the  ends  of  the  beam,  or  other  attendant  circumstances  of 
the  same  general  character.  Inserting  these  results  in 
eq.  (6),  the  expression  for  the  deflection  will  become 


Eq.  (6a)  is  not  often  used,  but  there  are  some  practical 
applications  of  formulae  in  which  it  must  be  employed, 


Art.  2;.]  DEFLECTION  DUE   TO  SHEARING.  125 

Deflection  Due  to  Shearing. 

That  portion  of  the  deflection  due  to  transverse  shear- 
ing may  be  determined  as  readily  as  that  due  to  the  length- 
ening and  shortening  of  the  fibres  of  the  bent  beam.  In 
determining  the  requisite  equations  it  is  necessary  to  con- 
sider only  the  intensity  of  shear  in  the  neutral  surface, 
as  it  is  the  deflection  of  that  surface  which  is  sought. 

Let  w'  be  the  deflection  due  to  shearing  and  let  <j>  repre- 
sent the  transverse  shearing  strain  for  a  unit  of  length  of 
the  beam.  The  transverse  strain  for  an  indefinitely  short 
portion  dx  of  the  neutral  surface  will  then  be  dw'  =  (/>dx, 
If  G  represents  the  coefficient  of  elasticity  for  shear,  while 
5  represents  the  intensity  of  shear,  eq.  (3)  of  Art.  2  shows 

that  <j>=7=;-     There  may  then  be  written: 

dwf  =  <f>dx  =  ~dx  .......     (7) 

By  using  the  value  of  5  given  in  eq.  (7)  of  Art.  i$> 

(8) 


The    general   expressions   for   the    shearing   deflection 
will,  therefore,  take  the  form: 


The  integration  required  in  eq.  (9)  can  be  made  with 
ease  in  any  given  case,  as  it  is  necessary  only  to  express 
the  value  of  the  total  transverse  shear  5  in  terms  of  x. 
The  application  of  that  equation  to  special  cases  will  be 


126  FLEXURE.  [Ch.  II. 

made  in  a  later  article.     Obviously  the  total  deflection  in 
any  bent  beam  will  be  the  sum : 

w  +  u/.' .      ......     (10) 


Art.  28.  —The  Neutral  Curve  for  Special  Cases. 

The  curved  intersection  of  the  neutral  surface  with  a 
vertical  plane  passing  through  the  axis  of  a  loaded,  and 
originally  straight,  beam  may  be  called  the  "neutral 
curve."  The  neutral  curve  is  the  locus  of  the  extremities 
of  the  ordinates  w  of  Art.  27;  it  therefore  gives  the  deflec- 
tion at  any  point  of  the  beam  due  to  the  direct  stresses  of 
tension  and  compression  in  it,  but  not  due  to  the  effect  of 
transverse  shear,  which  will  be  treated  in  a  subsequent 
article. 

The  method  of  finding  the  neutral  curve  for  any  par- 
ticular case  of  beam  or  loading  can  be  well  illustrated  by 
the  operations  in  the  following  three  cases: 

Case  I. 

This  case  is  shown  in  the  accompanying  figure,  which 
represents  a  cantilever  carrying  a  uniform  load  with  a 


---X- 


I 

FIG.  T. 

single  weight  W  at  its  free  end.     As  usual,  the  intensity 
of  the  uniform  loading  will  be  represented  by  p. 


Art.  28.]         THE  NEUTRAL   CURVE  FOR  SPECIAL   CASES.  127 

Measuring  x  and  w  from  B,  as  shown,  the  general  value 
of  the  bending  moment  is 


(I) 


Integrating  between  x  and  /,  remembering  that : 

dw 


for  x=l: 


Hence 

A    i   (W. 


The  greatest  deflection,  wx,  occurs  for  x  =  l.     Hence 


This  value  of  wl  is  the  deflection  of  B  below  A.  The 
general  value  of  w  in  eq.  (3)  is  the  vertical  distance  (de- 
flection) of  B  below  the  point  located  by  x ;  as  an  ordinate 
it  is  measured  upward  from  B  as  the  origin  of  coordinates. 

The  greatest  moment,  Mlt  exists  at  A,  and  its  value  is: 


(5) 


128  FLEXURE.  [Ch.  II. 

These  equations  are  made  applicable  to  a  cantilever 
with  a  uniform  load  by  simply  making  W  =o.  They  then 
become 


(6) 

(7) 


(10) 


Again,  for  a  cantilever  with  a  single  weight  only  at  its 
free  end,  p  is  to  be  made  equal  to  zero  in  the  first  set  of 
equations.  Those  equations  then  become  : 


,  ......   (n) 

dw     W 
f  =—  (*2-/2),       ......  (12) 

dx     2  ^ 


W 


(15) 


Art.  28.]         THE  NEUTRAL   CURYE  FOR  SPECIAL   CASES. 


129 


The  general  expressions  for  the  shear  and  the  intensity 
of  loading  are  : 


(16) 


(17) 


Case  II. 

This  case,  shown  in  the  figure,  is  that  of  a  non -continu- 
ous beam,   supported  at  each  end,  and  carrying  both  a 


4 x 


w 

FIG.  2 

uniform  load  (whose  intensity  is  p)  and  a  single  weight  W 
at  its  middle  point.  The  reaction  R,  at  either  end,  will 
then  be 

R_J*  +  W 


The  general  value  of  the  moment  will  then  be 


fi8) 


The  origin  of  x  and  w  is  taken  at  A. 
Remembering  that 

dw  I 

~T~=O     for     x——t 
dx  2 


fLEXURE.  [Ch.  II. 


and  integrating  between  the  limits  x  and  -, 


R(  ,     12\     pi  , 
- 


Again  integrating 

T    (  7?/r3      r/2\       -fr/r4      r/3N 
I    \  i\    x       %i  \      pix       %i   .  . 

<w=-^ri-\~ r)~A  (-— Q-Jr-     •    •    (2°) 


The  greatest  deflection  7e;t  occurs  at  the  centre  of  the 
span,  for  which 


I 

00  = 

Hence 


x=—. 

2 


The  greatest  moment,  also,  is  found  by  putting 


•#=-. 

2 


It  has  the  value 


These  formulae  are  made  applicable  to  a  non-continuous 
beam  carrying  a  uniform  load  only,  by  putting  W  =  o. 
They  then  become 

7?     Pl 
K  =  ^' 

tV-,).       ....     (23) 


Art.  28.  j         THE  NEUTRAL   CURVE  FOR  SPECIAL   CsiSES.  131 

dw          /xH      X3       Za 


-*<-/%)  .....  (25) 

5     Pi*  ,  ,. 

-8'48£/'   '     '     '     '  &6 

....  (27) 


The  formulas  for  a  beam  of  the  same  kind  carrying  a 
single  weight  at  the  centre  are  obtained  by  putting  p  =  o 
in  the  first  set  of  equations.  Those  for  the  greatest  deflec- 
tion and  greatest  moment,  only,  however,  will  be  given. 
They  are 

Wl3 

»>f&  xwi  •   (28) 


Wl 

........     (29) 


The  general  values  of  the  shear  and  intensity  of  loading 
are 


(30) 


"  dx  - 

d2M 
dx2=  ~p 

Case  III. 

The  general  treatment  of  continuous  beams  requires  the 
use  of  the  theorem  of  three  moments.  The  particular  case 
to  be  treated  is  shown  in  Fi£.  3.  The  beam  covers  the 


132  FLEXURE.  [Ch.  II. 

three  spans,  DA,  AB,  and  BC,  and  is  continuous  over  the 
two  points  of  support,  A  and  B. 

Let  DA  =1^ 
11    AB=12     Let/a  =«/!=»'/,. 


Let  the  intensity  of  the  uniform  load  on  AB  be  repre- 
sented by  p  and  let  the  two  single  forces  P  and  P'  only,  act 


.  0 

mmmc  

A                          _^J?                       C~^}           C 

FIG.  3. 

in  the  spans  DA  and  BC  respectively.  Also  let  the  two 
distances 

DE  =zl=  a/t     and     CF  =  a'/3 

be  given.  //  is  required  to  find  the  magnitudes  of  the  forces 
P  and  P',  if  the  beam  is  horizontal  at  A  and  B. 

Since  the  beam  is  horizontal  at  A  and  B,  the  bending 
moments  over  those  two  points  of  support  will  be  equal 
to  each  other,  for  the  load  on  AB  is  both  uniform  and 
symmetrical.  Let  this  bending  moment,  common  to  A 
and  B,  be  represented  by  M2.  As  the  ends  of  the  beam 
simply  rest  at  D  and  C,  the  moments  at  those  two  points 
reduce  to  zero. 

Because  the  four  points  D,  A,  B,  and  C  are  in  the  same 
level,  the  first  member  of  eq.  (14)  of  Art.  24  becomes  equal 
to  zero. 

If  that  equation  be  applied  to  the  three  points  D,  A, 


Art.  28.]        THE  NEUTRAL   CURVE  FOR  SPECIAL   CASES.  133 

and  B,  the  conditions  of  the  present  problem  produce  the 
following  results: 


and 


Hence  the  equation  itself  will  become 


=o.  .  .  (32) 

*-- 


,  ,    , 

•   '    (33) 


/.     Reaction  at  D=Rl=P±-  +      .  .     .     (34) 

l\         l\ 

As  the  origin  of  zl  is  at  D,  x  will  be  measured  from  the 
same  point. 

Separate  expressions  for  moments  must  be  obtained  for 
the  two  portions,  DE  and  EA  of  /p  because  the  law  of 
loading  in  that  span  is  not  continuous. 

Taking  moments  about  any  point  of  EA 


(35) 


Remembering  that 

dw 


134  FLEXURE.  [Ch.  II. 

for  x  =  lv  and  integrating  between  the  limits  x  and  /t 

EI^^x'-W-^W-l^+Pz^-lJ.   .     (36) 

Again,  remembering  that  w  =  o  for  x  —  lv  and  integrat- 
ing between  the  limits  x  and  Zp 


J.     (37) 
Taking  moments  about  any  point  in  DE 

-  .     (38) 


Making  x=z1  in  eqs.  (36)  and  (39),  then  subtracting 


^-^-     (40) 

Remembering  that  w  =  o  for  A;=O,  and  integrating  be- 
tween the  limits  x  and  o, 


^-lJx.      (41) 
Making  x=z±  in  eqs.  (37)  and  (41),  then  subtracting 


Art.  28.]         THE  NEUTRAL   CURYE  FOR  SPECIAL   CASES.  135 

Putting  the  value  of  M2  from  eq.  (33)  in  eq.  (34),  then 
inserting  the  value  of  Rv  thus  obtained,  in  eq.  (42),  after 
making  z^  =alv 


2+3W 
.       p_ 


2  'J        4(2  +3W)' 


6a(i-a2)     6a(i-a2)' 


This  is  the  desired  value  of  P,  which  will  cause  the 
beam  to  be  horizontal  over  the  two  points  of  support  A 
and  B  when  the  span  AB  carries  a  uniform  load  of  the 
intensity  p. 

By  the  aid  of  eq.  (43),  eq.  (33)  now  gives 

71  /[     _  >^7   2  ^  '    O    •   f  f  1  x    2  t         \ 

(44) 


,  . 

z  a  (9 +  3**)  12  12 

It  is  to  be  noticed  that  M2  is  entirely  independent  of 
t  or  13.     Eq.  (43)  also  gives 


Hence 

Thus  any  of  the  preceding  equations  may  be  expressed 
in  terms  of  p  or  P. 
Rl  also  becomes 

M  (47) 


or 

(48) 


136  FLEXURE.  [Ch.  II. 

It  is  clear  that  there  cannot  be  a  point  of  no  bending  in 
DE.  Hence  the  point  of  contra-flexure  must  lie  between 
E  and  A,  Fig.  3.  In  order  to  locate  this  point,  according 
to  the  principles  already  established,  the  second  member 
of  eq.  (35)  must  be  put  equal  to  zero.  Doing  so  and  solving 
for  x 


(49) 


Since  P  is  always  greater  than  Rv  there  will  always  be 
a  point  of  contra-flexure. 

All  these  equations  will  be  made  applicable  to  the  span 
BC  by  simply  writing  a'  for  a,  13  for  lv  and  n'  for  n. 

As  an  example,  let 

a=\     and     n  =  i. 
Eqs.  (43),  (44),  and  (47)  then  give 


=  _ 


12  1  6  ' 


after  writing, 


In  general,  the  span  /,  is  called  "  a  beam  fixed  at  one 
end,  simply  supported  at  the  other  and  loaded  at  any  point 
with  the  single  weight,  P." 

Let  it,  again,  be  required  to  find  an  intensity,  "  //,"  of  a 
uniform  load,  resting  on  the  span  lv  which  will  cause  the 
beam  to  be  horizontal  at  the  points  A  and  B. 


Art.  28.]          THE  NEUTRAL   CURVE  FOR  SPECIAL  CASES.  137 

Since  the  load  is  continuous,  only  one  set  of  equations 
will  be  required  for  the  span.  The  equation  of  moments 
will  be 


„  ,     . 

o  =R.x—L  —  ......     (co) 

dx2  2 

Integrating  between  the  limits  x  and  llt 

^-^<*'-l,')-£<*'-V>..     .    .     (si) 

Integrating  between  the  limits  x  and  o, 


But,  also,  w=o,  when  x--=lr     Hence 
^i3_£V 

^7  =  :~Y~;  •'•  R*~™1* (53) 

This  equation  gives  the  value  R1  when  pf  is  known. 
Making  x  =  l^  in  eq.  (50),  and  using  the  value  of  Rl  from 
eq-  (53). 

Adapting  eq.  (32)  to  the  present  case, 


4(2 


(     } 


Equating  these  two  values  of  Mv 

.......     (56) 


138  FLEXURE.  [Ch.  II. 

Thus  is  found  the  desired  value  of  pf.  In  this  case  the 
span  /t  is  called  "  a  beam  fixed  at  One  end,  simply  sup- 
ported at  the  other  and  uniformly  loaded." 

The  points  of  contra  -flexure  are  found  by  putting  the 
second  member  of  eq.  (50)  equal  to  zero  and  solving  for 
x,  after  introducing  the  value  of  Rt  from  eq.  (53).  Hence 


or 

oc  =  o     and     x  =  $lr 

Between  the  simply  supported  end  and  point  of  contra- 
flexure  the  beam  is  evidently  convex  downward,  and  convex 
upward  in  the  other  portion  of  the  spans  /x  and  /3,  whether 
the  load  is  single  or  continuous.  Moments  of  different 
signs  will  then  be  found  in  these  two  portions,  and  there 
will  be  a  maximum  for  each  sign.  The  location  of  the 
sections  in  which  these  greatest  moments  act  may  be  made 
in  the  ordinary  manner  by  the  use  of  the  differential  cal- 
culus; but  the  negative  maximum  is  evidently  M2,  given 
by  eqs.  (44)  and  (55).  On  the  other  hand,  the  positive 
maximum  is  clearly  found  at  the  point  of  application  of 
P  in  the  case  of  a  single  load,  and  at  the  point 

x  —  3.7 

•^  —  s^i 

in  the  case  of  a  continuous  load.  These  conclusions  will  at 
once  be  evident  if  it  be  remembered  that  the  portion  of  the 
beam  between  the  supported  end  and  point  of  contra  - 
flexure  is,  in  reality,  a  beam  simply  supported  at  each  end. 
These  moments  will  have  the  values 


(57) 
(58) 


Art.  28.]         THE  NEUTRAL   CURVE  FOR  SPECIAL   CA-SES.  139 

In  case  of  a  single  load  if  P  is  given,  and  not  p,  e"q.  (45) 
shows 


The  points  of  greatest  deflection  are  found  by  putting 
the  second  members  of  eqs.  (36),  (40),  and  (51)  each  equal 
to  zero,  and  then  solving  for  x.  They  are  not  points  of 
great  importance,  and  the  solutions  will  not  be  made. 

The  following  are  the  general  values  of  the  shears  for  a 
single  load  on  /x: 

InAE,     S=El-j^=Ri-P\      [from  eq.  (35)]. 

In  ED     S,  =  El~^  =R1\  [from  eq.  (38)]. 

The  shear  in  ^  for  the  uniform  load  p'  is 

••Rl  —  pfx',   [fromeq.  (50)]. 


Also 


Intensity  of  load  =  El  -7-;  =  —  //. 


As  has  already  been  observed,  all  the  equations  relating 
to  the  span  /t  may  be  made  applicable  to  the  span  /3  by 
changing  a  to  a'  and  n  to  n' '. 

The  span  12  remains  to  be  considered. 

Since  the  bending  moments  at  A  and  B  are  equal  to 
each  other,  and  since  the  loading  is  uniformly  continuous, 
half  of  it  (the  load  pl2)  will  be  supported  at  A  and  the  other 
half  at  B.  In  other  words,  the  vertical  shear  at  an  in- 
definitely short  distance  to  the  right  of  A,  also  to  the  left 


HO  FLEXURE.  [Ch.  II. 

of  B,  will  be  equal  to  —  .     Let  x  be  measured  to  the  right 
and  from  A.     The  bending  moment  at  any  section  x  will  be 
d*w  pl2       px2 

or 

/y/y  Q,  \<DZ7/ 

Integrating  between  the  limits  x  and  o, 

El  -j-  =  MJX  +—  ( — ) .  (60) 

dx  2  \  2        3  / 

Again,  integrating  between  the  same  limits, 


Since 

dw 

dx  ~° 

for  x  =  lv  eq.  (60)  wi.l  give  M2  independently  of  preceding 
equations.     Following  this  method,  therefore, 


12 

This  is  the  same  value  which  has  already  been  obtained. 
Introducing  the  value  of  M2, 


w.dw    ptlj?     x*     /22  \ 
El  -r  --(-*—  —  —  --x},    .  (67) 

dx     2\  2  6    /  v  6J 


=^---.  (64) 

12  \  2  2  2 


Art.  28.]         THE  NEUTRAL   CURVE  FOR  SPECIAL   CASES,  141 

The  points  of  contra-flexure  are  found  by  putting  the 
second  member  of  eq.  (62)  equal  to  zero.     Hence 

'      x*-lx--V> 
***          6  ' 

0.789/2. 

The  moment  at  the  centre  of  the  span  is  found   by 
putting 


ineq.  (62): 


24 


This  is  the  greatest  positive  moment 
The  general  value  of  the  shear  is 


,T 
S=EI 


3 
dx* 

and  the  intensity  of  load 


The  span  /,  is  generally  called  "  a  beam  fixed  at  both 
ends  and  uniformly  loaded." 

It  is  sometimes  convenient  to  consider  a  single  load  at 
the  centre  of  the  span  /2,  while  the  beam  remains  horizontal 
at  A  and  B\  in  other  words,  to  consider  "  a  beam  fixed  at 
each  end  and  supporting  a  weight  at  the  centre." 

Let  W  represent  this  weight;  then  a  half  of  it  will  be 
the  shear  at  an  indefinitely  short  distance  to  the  right  of 


142  FLEXURE.  [Ch.  II. 

A  and  left  of  B.    As  before,  let  %  be  measured  from  A,  and 
positive  to  the  right.     The  moment  at  any  point  will  be 


(65) 


'dx2 
Integrating  between- #  and  o, 

^=M2*-— 2 (66) 

dx  4 

If  x=— ,  then  will 
2 

dw  _ 
dx 

hence  Mz  =  —A 

o 

The  general  value  of  the  moment  then  becomes 
d2w      WL     Wx 


If  x=-  in  this  equation,  the  bending  moment  at  the 
centre  (where  W  is  applied)  has  the  value 

Wl 

Centre  moment  =  --  r-1  . 

o 

Hence  the  bending  moments  at  the  centre  and  ends'  are  each 
equal  to  the  product  of  the  load  by  one  eighth  the  span,  but 
have  opposite  signs. 

A  second  integration  between  x  and  o  gives 


Hence  the  deflection  at  the  centre  has  the  value 

Wl  3 
Centre  deflection  =        " 

*  The  use  of  the  signs  in  this  and  the  following  equations  is  changed 
from  the  preceding  to  show  that  either  procedure  may  be  employed. 


Art.  28.] 


THE  NEUTRAL   CURVE  FOR  SPECIAL   CASES. 


143 


By  placing  M  =  o,  the  points  of  contra-flexure  are  found 
at  the  distance  from  each  end, 


Addendum  to  Art.  28. 

The  formulae  of  this  article  furnish  the  solutions  of  many 
practical  questions  of  maxima  deflections  and  moments. 
The  latter  for  several  ordinary  cases  are  given  in  the  follow- 
ing tabulation: 

P  is  the  weight  in  pounds  at  end  of  beam  or  centre  of  span. 

p  is  the  load  in  pounds  per  lin.  ft.  of  beam. 


Beam. 

Maximum 
Moment. 

Maximum 
Deflection. 

Point  of 
Contra-flexure  . 

I 

^^ 

PI  at  A 

PI3 

576^  at  A 

A 

II 

of  length. 

%pP  at  A 

216^  at  A 

III 

IV 

A 

\Pl  at  centre 
^pP  at  centre 

PI3 

A 

22.  ^r  at  centre 

^P  per  unit  W^ 
^  ~of  length.    ^ 

V 

jpr=|r 

-&PI  at  A 
^Pl  at  centre 

El  from  B 

T8T/  from  B 
Reaction  at  B 

^^^      B 

T5 

VI 

of  length. 

from  B 

El  from  B 

11  from  B 
Reaction  at  B 

VII 

4^ 

-\PlatA 
%Pl  at  centre 

PI3 
9gj  at  centre 

\l  from  each  end 

VIII 

-sA                ^ 
of  length. 

~T2pP  at  A 
-g^pP  at  centre 

pi* 
4.  51—  at  centre 

o.  2  1  1  /  from  each 
end 

144  FLEXURE.  [Ch.  II. 

/  is  the  length  of  beam  or  of  span  in  feet. 

E  is  the  coefficient  of  elasticity  in  pounds  per  sq.  inch. 

/  is  the  moment  of  inertia  of  the  normal  section  of  the 
beam  with  all  dimensions  of  section  in  inches. 

The  "  Max.  Moments"  will  be  in  foot  pounds,  and  the 
"  Max.  Deflections  "  will  be  in  inches. 

In  the  use  of  flexure  formulae,  in  many  practical  appli- 
cations, it  is  best  to  have  the  moment  M  in  inch-pounds, 
which  will  result  from  simply  multiplying  the  "  Max. 
Moments  "  of  the  preceding  table  by  12. 

Case  I  results  from  eqs.  (14)  and  (15);  Case  II  from 
eqs.  (9)  and  (10);  Case  III  from  eqs.  (28)  and  (29); 
Case  IV  from  eqs.  (26)  and  (27).  In  Case  V  the  reaction 
is  found  by  putting  a  =  J  in  eq.  (48);  the  point  of  "  Max. 
Deflection"  is  found  by  placing  zl  =  J/  in  eq.  (40),  and  the 

resulting  value  of  -7-  equal  to  zero  and  solving  for  %,  which 
d% 

latter  value  in  eq.  (41)  will  give  "  Max.  Deflection." 
Case  VI  results  from  treating  eqs.  (53),  (51),  and  (52)  in 
precisely  the  same  manner.  Case  VII  results  directly 
from  the  formulas  on  page  142.  Case  VIII  results  directly 
from  the  equations  on  pages  140  and  141. 

The  preceding  cases  are  those  which  commonly  occur 
with  constant  values  of  E  and  I.  Other  cases,  such  as  a 
single  load  at  any  point,  or  partial  uniform  load  over  any 
part  of  span,  are  to  be  treated  by  the  same  general  prin- 
ciples. 

Art.  29. — Direct  Demonstration  for  Beam  Fixed  at  One  End 
and  Simply  Supported  at  the  Other  Under  Uniform  and 
Single  Loads. 

A  beam  is  said  to  be  fixed  at  one  end  when  it  is  under 
such  constraint  that  the  neutral  surface  does  not  change 
its  direction  at  that  end  whatever  may  be  the  loading. 


Art.  29.]     DEMONSTRATION  FOR  BEAM  FIXED  AT  ONE  END.          145 

This  fixedness,  as  has  been  fully  shown  in  Art.  28,  is  equiv- 
alent to  the  application  of  a  suitable  constraining  moment. 
Beams  with  one  or  both  ends  under  such  constraint  have 
been  fully  treated  in  Art.  28,  but  it  is  desirable  to  establish 
the  formulae  for  such  cases  directly,  i.e.,  without  the  employ- 
ment of  the  theorem  of  three  moments. 

In  Fig.  i  a  beam  is  shown  fixed  at  one  end  B  and  simply 
supported  at  the  other  end  A,  while  it  carries  a  uniform 
load  py  per  linear  unit  and  the  single  load  P  at  the  distance 
al  from  A.  The  length  of  span  is  /  and  the  coordinate  % 


FIG.  i, 


is  measured  horizontally  to  the  right  from  A.  The  two 
reactions  are  R  and  R' .  E  is  the  modulus  of  elasticity, 
/  the  moment  of  inertia  of  the  normal  section  of  the  beam, 
and  w  is  the  deflection  at  any  point.  The  bending  moment 
for  any  point  in  the  segment  al  of  the  beam  is : 


The  bending  moment  for  the  section  l—alof.  the  beam  is 


Integrating  eq.  (i)  and  representing  by  C  the  constant 
of  integration: 

^7« /i  •  /y-2i  /y*3 

(3) 


dx 


146    .  FLEXURE.  [Ch.  II. 

Integrating  eq.  (3)  between  %  and  o,  remembering  that 
w  =  o,  when  x=o; 


(4) 
o       24 

Integrating  eq.  (2)  between  x  and  /,  remembering  that 


dw  -  7 

—  =  o  when  x  =  /, 
dx 


.     (5) 


If  #  =  a/  in  eqs.  (3)  and  (5),  the  first  members  of  those 
equations  will  be  equal,  hence  : 

n2l2        -/73/3 
U  i>         -a  i 


Taking  the  difference  between  (6)  and  (7)  and  solving 
forC: 


Placing  this  value  of  C  in  eq.  (4)  : 

EIw=-(x*  -3/2*)  —  £(**  -4fe)4.(a  _  l)2^          (  ) 

O  24  2 

Integrating  eq.  (5)  between  the  limits  of  x  and  /: 


24 

-0a)'  do) 


Art.  29.]     DEMONSTRATION  FOR  BEAM  FIXED  AT  ONE  END.         147 

Making  oc  =  al  in  eqs.  (9)  and  (10)  and  subtracting  the 
former  from  the  latter,  there  will  result  : 


(n) 

This  equation  gives  the  reaction  required  to  enable  any 
of  the  preceding  formulae  to  be  applied  to  actual  compu- 
tations. The  loads  P  and  p,  as  well  as  the  quantity  a  are 
obviously  known  for  any  particular  case  or  problem.  With 
the  value  of  the  reaction  now  established  by  eq.  (n)  the 
deflection  or  the  tangent  of  inclination  of  the  neutral  sur- 

face —  may  be  at  once  computed  for  any  point  in  either 

part  of  the  beam.  The  fixing  or  constraining  moment 
required  to  keep  the  beam  horizontal  at  B  can  be  at  once 
determined  by  making  %  =  l  in  eq.  (2)  and  it  has  the  value; 


(12) 


If  the  load  is  wholly  uniform  or  P=o,  eqs.  (n)  and  (i) 
give: 

R=$pl  andM=  -2- (13) 

o 

This  value  of  M  is  the  constraining  moment  required 
at  B  when  the  load  is  wholly  uniform  and  is  identical  with 
eq.  54  of  Art.  28.  Indeed  the  preceding  equations  are  the 
same  as  those  established  for  the  continuous  span,  con- 
ditioned similarly  to  the  beam  treated  in  this  article. 

In  all  the  preceding  equations  if  the  load  is  wholly 
uniform  it  is  only  necessary  to  make  P  =o.  On  the  other 
hand,  if  there  is  a  single  load  with  no  uniform  loading 


I4§  FLEXURE.  [Ch.  II. 

Inasmuch  as  the  beam  is  convex  downward  over  its 
left-hand  part  and  convex  upward  in  the  vicinity  of  B, 
there  must  be  a  point  of  contraflexure  either  to  the  right 
or  to  the  left  of  P,  according  to  its  location.  If  that  point 
is  between  P  and  B,  the  second  member  of  eq.  (2)  must 
be  placed  equal  to  o,  giving; 


,  2(P-R)         Pal 
* 


, 
Solving  this  quadratic  equation; 


Eq.  (15)  gives  the  location  of  the  point  of  contra- 
flexure by  the  value  of  %  measured  from  A.  There  are 
two  roots  of  the  equation,  but  evidently  the  positive  value 
of  the  radical  only  is  required. 

If  the  point  of  contraflexure  is  between  P  and  L4,  which 
would  be  the  case  if  the  single  load  were  near  the  right- 
hand  end  of  the  span,  the  second  member  of  eq.  (i)  must 
be  placed  equal  to  o,  giving; 


In  case  the  point  of  contraflexure  is  at  the  point  of 
application  of  P,  x  =  al,  hence, 

2R  2R  f       N 

00=  —  =  aZanda=  —  r  .....     (17) 

P  PL 

If  it  is  desired  to  find  the  point  at  which  the  deflection 

is  geratet,  it  is  osnly  necessary  to  place  -^  =  o  in  either 

doc 


Art.  29.]     DEMONSTRATION  FOR  BEAM  FIXED  AT  ONE  END.         149 

eqs.  (3)  or  (5),  as  the  case  may  be,  and  solve  the  resulting 
eq.  for  oc. 

The  reaction  R,  i.e.,  the  end  shear  at  B,  is; 

R'=pl+P-R  ......     (X8) 


The  sum  of  the  two  reactions  must  be  equal  to  the 
total  load  on  the  beam. 

Special  Case,  a=J. 

In  this  case  eq.  (n)  will  give  the  reaction  R  at  A  as 
follows:  . 


.    •  >    •    ,     (19) 
Hence,  the  reaction  Rr  at  B  will  be: 

£P..     ...     (20) 


The   fixing   or   constraining   moment   Mi   at   B   is   by 
eq.  (12): 


Eq.  (15)  shows  that  the  position  of  the  point  of  contra- 
flexure  will  depend  upon  the  magnitude  of  P.  If  P=o 
that  equation  shows  that  the  point  of  contraflexure  will 
be  {/  from  A  : 

*  =  |/  .......     (22) 

The  part  \l  of  the  span  will  be  in  the  condition  of  a 
beam  simply  supported  at  each  end  and  uniformly  loaded. 
Hence  the  greatest  positive  bending  moment  at  the  dis- 
tance |/  from  A  is: 

(23) 


FLEXURE. 


[Ch.  II. 


The  point  of  greatest  deflection  will  be  found  by  placing 
the  second  member  of  eq.  (5)  =o  and  solving  for  x. 


Art.  30. — Direct  Demonstration  for  Beams  Fixed  at  Both  Ends 
under  Uniform  and  Single  Loads. 

Fig.  i  shows  a  horizontal  beam  with  both  ends  fixed, 
so  that  whatever  may  be  the  magnitudes  of  the  uniform 
loading  and  the  single  load,  or  the  position  of  the  latter, 
the  neutral  surface  at  each  end  of  the  beam  remains  hori- 


zontal. The  coordinate  x  is  measured  from  the  left-hand 
end  A  of  the  beam  as  is  also  the  distance  al  of  the  single 
load  P  from  the  left  end  of  the  span.  The  length  of  the 
span  is  /  and  the  reactions  or  shears  at  the  ends  of  the 
span  are  indicated  by  R  and  R' .  The  fixing  or  constrain- 
ing moment  at  A  is  indicated  by  MQ  and  the  uniform  load 
per  linear  unit  by  p.  If  as  before  w  represents  the  deflec- 
tion at  any  point,  the  equation  of  moments  for  the  part 
al  of  the  beam  may  at  once  be  written : 


(i) 


Integrating  eq.    (i)   between  the  limits   x  and  o,   and 
remembering  that  —  =  o  f or  x  =  o ; 


dw 
~dx~ 


px3 

6  ' 


(2) 


Art.  30.]  DEMONSTRATION  FOR  BEAMS  FIXED  AT  BOTH  ENDS.     151 

Integrating  eq.  (2)  between  the  limits  x  and  o,  eq.  (3) 
may  be  at  once  written  as  w  =  o  f  or  oc  =  o  : 

EIw=M<£+Rj-p*  .....     (3) 

Proceeding  in  the  same  manner  for  that  part  of  the 
beam  between  B  and  the  load  P  the  equation  of  moments  is  : 

EI^=M0+Rx-p--P(x-al).      I    ,.     (4) 

ax2  2 

Integrating  eq.  (4)  between  the  limits  of  x  and  /,  since 

dw        c  j 

-*-  =  o  for  x  =  I  ; 

ax 

El  ^  =M0(>2  -/2)  +-(x2-P)  -£(*3  _/3) 
ax  26 


Again  integrating  between  the  limits  x  and  /: 


24 

(6) 


2\3 

The  two  unknown  quantities  M0  and  R  are  to  be  found. 
By  placing  x  —  al  in  eqs.  (2)  and  (5),  then  subtracting  the 
former  from  the  latter  : 

o=-M0--/+^  +  -(a-i)2.      ...     (7) 

202 

Again  making  x=alm  eqs.  (3)  and  (4),  then  subtracting 
the  former  from  the  latter; 

Mox  \     Rh          v  i  P12(          \  ,  Ph  /0\ 

o  =  ---  (20  -  1)  —  —  (30  -  2)  +^—  (4^-3)  +—  (a  -  1)3.        (8) 
2  6  24  3 


152  FLEXURE.  [Ch.  II. 

If  eq.  (7)  be  multiplied  by  \(ia  —  i)  and  then  subtracted 
from  eq.  (8),  the  following  value  of  the  reaction  or  end 
shear  will  at  once  result  : 


(9) 


By  placing  this  value  of  R  in  eq.  (7),  the  value  of  Mo 
at  once  follows  : 


i)2.  (10) 

12 

In  order  to  determine  the  moment  Mi  at  the  end  B 
of  the  span,  it  is  only  necessary  to  substitute  the  preceding 
values  of  Mo  and  R  in  eq.  (4)  : 


i-a)  .....    .     .     (n) 

These  equations  give  all  the  quantities  required  for  the 
complete  solution  of  the  case.  The  reaction  or  end  shear 
at  B  is  simply  : 

pl+P-R  =  ^+P(i-(a-i)2(2a  +  i))    .     .     (12) 

The  greatest  negative  bending  moment  will  obviously 
be  found  at  either  one  end  or  the  other  of  the  span,  depend- 
ing upon  the  value  of  a  c,nd  the  amount  of  the  load  P. 
The  greatest  positive  bending  moment  will  be  found  where 
the  shear  is  zero. 

There  will  be  two  points  of  contraflexure,  one  in  each 
segment  of  the  span.  That  point  located  in  the  part  al 
will  be  determined  in  the  usual  manner  by  placing  the 
second  member  of  eq.  (i)  equal  to  zero  and  solving  the 
quadratic  equation.  This  simple  operation  will  give  eq. 

da):  _ 

R      l2M0    & 


Art.  30.]  DEMONSTRATION  FOR  BEAMS  FIXED  AT  BOTH  ENDS.    153 

Proceeding  in  the  same  manner  with  eq.  (4)  there  will 
result : 


(M+Pa 


This  last  value  of  %  will  indicate  the  point  of  contra- 
flexure  for  the  right-hand  part  of  the  beam. 

Special  Case,  a=|. 

If  P  be  placed  at  the  center  of  the  span,  a  =  J  and  eqs. 
(9),  (10),  and  (n)  will  give  eq.  (15): 


.    .     . 

22  12  8 

The  moment  M1  at  the  centre  of  the  span  will  be  given 
by  the  aid  of  eq.  (i): 


24        o 


The  greatest  deflection  w\  is  at  the  centre  of  the  span 

and  it  is  given  by  placing  %  =—  in  eq.  (3). 

2 

Rl 


The  values  of  Mo  and  R  are  given  by  eq.  (15). 

Art.  31.  —  Deflection  Due  to  Shearing  in  Special  Cases. 

The  deflection  due  to  transverse  shearing  only  in  all 
the  ordinary  cases  of  loaded  beams  can  readily  be  com- 
puted by  aid  of  the  general  eq.  (9)  of  Art.  27.  If  d  is 
the  distance  from  the  most  remote  fibre  from  the  neu- 


154 


FLEXURE. 


[Ch.  II. 


tral  axis  of  any  normal  section  whose  moment  of  inertia 
about  the  same  axis  is  /,  and  if  G  and  5  are  the  coefficient 
of  elasticity  and  total  transverse  shear  respectively,  the 
deflection,  w',  sought  is 


-J 


r^,  /  Sd*. 


(I) 


The  limits  of  the   integration  must  be   indicated  for 
each  particular  case. 


FIG.  i. 


In  Fig.  i  let  the  cantilever,  whose  length  is  /,  carry  the 
single  load  P  at  its  end,  and  the  uniform  load  p  per 
linear  unit.  The  shear  at  any  section  distant  x  from  A  is 


S=P-\-px. 
will  give 


The  substitution  of  this  value  of  5  in  eq.  (i) 


wf  =• 


f 

*J  0 


(2) 


If  the  uniform  load  only  acts,  P  =o ;  and  if  P  only  acts, 

Fig.  2  shows  the  case  of  a  simple  beam  supported  at 
each  end,  carrying  a  uniform  load  p  per  linear  unit  and 
the  single  load  P  at  the  centre  of  the  span.  The  reaction 
R  =  %(P  +  pl),  and  the  shear  S=R  —  px.  Hence  eq.  (i) 
gives  the  general  value  of  the  deflection 

d2      Cx  d2    ( %  p%* ) 

2lG  J  n  2lG  I   2  2      \  '  ^ 


Art.  31.]    DEFLECTION  DUE  TO  SHEARING  IN  SPECIAL  CASES.    155 
And  for  the  centre  ot   the  span: 


R 

!    (j) 

,R' 

ilillilP  :  "  x        t                                   •  ':'^PIPili 

1   , 

J                           J                                                  |f 

<;-L                               .    i  —                                 __R 

FIG.  2. 


The  values  of  the  deflection  w'  may  be  s_milarly  written 
for  other  cases.  The  following  table  gives  the  results  for 
the  cases  indicated,  which  are  those  commonly  required. 


Beam. 


End  Shear. 


Shear  S. 


Section  for 
Deflection  w'  . 


Deflection  w'. 


I 
II 
III 

IV 


px 


per  unit 
of  length. 


VI 

VII 
VIII 


/          from  end 


V 


.447/      " 


.4215*  " 


BIG 


i6IG 

d*Pl 
I4.32/G 


I2.8/G 


/G 


S/G 


i6/G 


156  FLEXURE.  [Ch.  II. 

The  end  shears  in  this  table  are  the  reactions  taken 
from  the  table  of  the  preceding  article,  the  "Beams"  in 
the  two  tables  being  the  same. 

The  total  deflection  for  any  particular  beam  is  to  be 
found  by  adding  the  "Max.  Deflection"  from  the  table 
of  the  preceding  article  to  the  w'  found  in  the  above  table. 

In  the  notation  of  the  preceding  article,  if  w1  is  the 
deflection  due  to  the  lengthening  and  shortening  of  the 
fibres  the  total  deflection  in  any  case  will  be 

w=w1+w'.       ......     (5) 

These  formulae  for  shearing  deflection,  like  all  the 
formulae  relating  to  the  distribution  of  transverse  shearing 
in  a  bent  beam,  are  more  accurately  applicable  to  rectan- 
gular or  circular  sections  than  to  others. 


Art.  32. — The  Common  Theory  of  Flexure  for  a  Beam  Composed 
of  Two  Materials. 

The  common  theory  of  flexure  as  set  forth  in  the  pre- 
ceding articles  is  applicable  to  a  beam  composed  of  two 
or  more  materials  with  minor  changes  only  in  the  formulas 
established,  but  two  different  materials  only  will  be  con- 
sidered here,  as  that  number  are  frequently  used  in  engi- 
neering works. 

Two  such  materials,  concrete  and  steel,  are  widely  used 
in  reinforced  concrete  beams.  Let  E  be  the  modulus  of 
elasticity  for  steel  and  E\  for  concrete,  and  let  e  represent 

the  ratio  between  the  two  moduli,  i.e.,  e= — .     This  ratio 

EI 

for  concrete  and  steel  is  generally  taken  as  15,  although 
12  is  sometimes  used.  Let  A  be  the  area  of  that  part  of 
the  section  with  the  modulus  E  and  Ait  the  area  of  section 


Art.  32.]  THE  COMMON   THEORY  OF  FLEXURE.  157 

having  the  modulus  E\.   Iiu=---  (the  reciprocal  of  the  radius 

P 

of  curvature)  be  the  strain  of  a  unit  length  of  fibre  at  unit 
distance  from  the  neutral  axis,  then  will  the  intensities 
of  the  direct  stresses  of  tension  or  compression  at  the  dis- 
tance z  from  the  neutral  surface  be : 

Ni=Eiuz  and  N=Euz=eE\uz. 

Inasmuch  as  the  two  materials  are  supposed  to  act  together 
as  a  unit,  the  rate  of  strain  will  be  the  same  for  both  at  a 
given  distance  from  the  neutral  axis. 

The  amounts  of  direct  stress  on  the  two  differential 
areas  dA\  and  dA  will  be  as  follows: 

EiuzdAi+EuzdA=aizdAi+eaizdA.       .     .  (i) 

a\  and  no\  are  intensities  of  stsess  at  unit  distance  from  the 
nutral  axis. 

The  sum  of  the  direct  stresses  of  tension  and  compression 
in  any  normal  section  of  the  beam,  if  the  beam  is  hori- 
zontal and  all  loading  vertical,  will  be  zero.  Hence: 

fzdAi+fessdA-o (2) 

The  limits  of  the  integrations  indicated  will  depend  upon 
the  form  of  cross-section  and  the  distribution  of  the  two 
materials.  Frequently  the  section  of  one  material,  such 
as  the  steel  in  reinforced  concrete  work,  is  but  a  small  per- 
centage of  the  total  cross-section,  and  it  is  sufficiently 
accurate  to  consider  it  concentrated  at  the  distance  d2  from 
the  neutral  axis  on  one  side  of  the  latter  and  at  the  dis- 
tance d3  from  the  same  axis  on  the  opposite  side.  If  d2 
is  considered  positive,  d%  must  be  taken  as  negative. 


i58  FLEXURE.  [Ch.  II. 

Finally,  if  A  2  and  A  3  be  taken  as  the  small  areas  of  section 
of  the  material,  eq.  (2)  will  take  the  form  of  eq.  (3) : 

=o (3) 

Invariably  the  small  sections  A2  and  A  3  belong  to  a 
material  with  a  far  higher  modulus  than  the  other.  In 
reinforced  concrete  the  sum  of  A 2  and  AS  is  usually  about 
i  per  cent  or  less  of  the  entire  cross-sectional  area  of  the 
beam  with  E  =30,000,000  and  E\  =2,000,000. 

When  the  form  of  cross-section  of  the  beam,  i.e.,  the 
cross-section  of  both  materials  of  which  the  beam  is  com- 
posed, is  known,  the  position  of  the  neutral  axis  of  the 
section  can  at  once  be  found  by  either  eq.  (2)  or  eq.  (3). 
It  is  obvious  from  these  equations  that  the  neutral  axis 
will  not  pass  through  the  centre  of  gravity  of  the  section. 
Whether  it  will  be  at  one  side  or  the  other  of  that  point  will 
depend  upon  the  amount  and  distribution  of  the  materials 
and  the  greater  modulus  of  elasticity. 

Frequently  the  steel  is  omitted  on  the  compression  side 
of  reinforced  concrete  beams  and  in  such  case  either  A2  or 
As  will  be  zero. 

The  bending  or  resisting  moment  of  the  internal  stresses 
in  any  normal  section  of  a  beam  can  be  written  at  once 
by  the  aid  of  the  second  member  of  eq.  (i).  If  that  second 
member  be  multiplied  by  z,  the  differential  resisting  moment 
will  at  once  result.  Hence: 

(4) 

As  indicated  in  eq.  (i),  a\  is  the  intensity  of  the  direct 
stress  of  either  tension  or  compression  in  a  fibre  at  unit 
distance  from  the  neutral  axis  for  the  material  with  the 
modulus  Ei.  The  integrals  in  eq.  (4)  will  be  recognized 


Art.  32.]  THE  COMMON    THEORY  OF  FLEXURE.  159 

at  once  as  the  moments  of  inertia  of  the  cross-section  of 
the  two  different  materials  about  the  neutral  axis  established 
by  eq.  (2)  or  eq.  (3).  If  the  same  assumptions  made  in 
connection  with  eq.  (3)  are  known  in  connection  with  eq. 
4  this  latter  equation  will  take  the  following  form : 

M=aifz2dA}+eai(A2d22+A3d23).       .     .     (5) 

Again  since  a\  =  —  =  --  =-^  eq.  (5)  may  take  the  follow- 
di     d2     dz 

ing  form : 


It  is  to  be  observed  that  k,  k2  and  kz  are  intensities  of 
stress  at  the  distances  from  the  neutral  axis  indicated  by 
d\,  d2,  and  d%  in  the  material  whose  modulus  of  elasticity 
is  Ei. 

These  equations  indicate  completely  the  only  modi- 
fications to  be  made  in  the  common  theory  of  flexure  as 
applied  to  one  material  for  a  beam  composed  of  two  differ- 
ent materials,  and  they  indicate  also  the  corresponding 
changes  necessary  to  adapt  the  common  theory  of  flexure 
to  a  beam  composed  of  more  than  two  different  materials. 

In  eqs.  (4),  (5),  and  (6)  the  moment  M  is  simply  the 
ordinary  expression  for  the  external  bending  moment  to 
which  a  beam  is  subjected  in  terms  of  the  horizontal  co- 
ordinate x  and  given  loads. 

The  formulae  to  be  used  to  compute  the  deflection  of 
a  beam  composed  of  two  materials  are  readily  written  by 
means  of  the  preceding  equations.  As 


k     k2  7-        EI      ^  d2w 

ai=— =— =etc.  =Eiu= —  =  £1-7-, 
di     d2  p 


eq.  (6)  gives: 


160  FLEXURE.  [Ch.  II. 

M  .....     (7) 


As  already  explained,  M,  the  external  bending  moment, 
is  expressed  in  terms  of  the  loads  and  the  coordinate  x. 
Eq.  (7)  therefore  can  be  integrated  precisely  as  in  the  case 
of  a  beam  of  a  single  material.  Indeed  there  is  no  differ- 
ence between  the  two  cases  except  that  istead  of  the 
moment  of  inertia  I  for  a  single  material,  the  term  I  +0/2  + 
elz  must  take  its  place,  the  latter  expression  being  the  sum 
of  the  three  components  of  the  resultant  moment  of  inertia 
of  the  combined  normal  section. 

The  first  integration  of  eq.  (7)  will  obviously  give  the 
tangent  of  the  inclination  of  the  neutral  surface  at  any 
point,  while  the  second  will  give  the  deflection. 

Art.  33.  —  Graphical  Determination  of  the  Resistance  of  a  Beam. 

The  graphical  method  is  well  adapted  to  the  treatment 
of  beams  whose  normal  sections  are  limited  either  wholly 
or  in  part  by  irregular  curves.  In  Fig.  i  is  represented 
the  normal  section  of  such  a  beam,  the  centre  of  gravity 
of  the  section  being  situated  at  C.  The  lines  HL,  AB, 
and  DF  are  parallel.  As  is  known  by  the  common  theory 
of  flexure,  the  neutral  axis  will  pass  through  C. 

Let  aa  be  any  line  on  either  side  of  AB,  then  draw  the 
lines  aa'  normal  to  AB,  having  made  MN  and  HL  equidis- 
tant from  AB.  From  the  points  a'  thus  determined  draw 
straight  lines  to  C.  These  last  lines  will  include  intercepts, 
bb,  on  the  original  lines  aa.  Let  every  linear  element 
parallel  to  AB,  on  each  side  of  C,  be  similarly  treated.  All 
the  intercepts  found  in  this  manner  will  compose  the  shaded 
figure. 

This   operation   in    reality,    and    only,    determines   an 


Art.  33.]         GRAPHICAL    METHODS   APPLIED  TO  BEAMS. 


161 


amount  of  stress  with  a  uniform  intensity  identical  with 
that  developed  in  the  layer  of  fibres  farthest  from  the 
neutral  axis,  and  equal  to  the  total  bending  stress  existing 
in  the  section ;  this  latter  stress,  of  course,  having  a  varia- 
ble intensity.  HL  represents  the  layer  of  fibres  farthest 
from  the  neutral  surface,  consequently  MN  was  taken  at 
the  same  distance  from  AB.  Any  other  distance  might 
have  been  taken,  but  the  intensity  of  the  uniform  stress 


of      N 


FIG.  i 

would  then  have  had  a  value  equal  to  that  which  exists 
at  that  distance  from  the  neutral  axis.  Again,  a  different 
intensity  might  have  been  chosen  for  the  stress  on  each 
side  of  AB.  It  is  most  convenient,  however,  to  use  the 
greatest  intensity  in  the  section  for  the  stress  on  both  sides 
of  the  neutral  axis ;  this  intensity,  which  is  the  modulus  of 
rupture  by  bending,  will  be  represented,  as  heretofore,  by  K. 
Let  c  and  cf  be  the  centres  of  gravity  of  the  two  shaded 
figures.  These  centres  can  readily  and  accurately  be  found 
by  cutting  the  figures  out  of  stiff  manilla  paper  and  then 
balancing  on  a  knife-edge.  Let  5  represent  the  area  of  the 


1 62  FLEXURE.  [Ch.  II. 

shaded  surface  below  A B,  and  s'  the  area  of  that  above 
AB. 

Because  this  is  a  case  of  pure  bending,  the  stresses  of 
tension  must  be  equal  to  those  of  compression.  Hence 

Ks=Ksr,     or     s=sf («) 

The  moment  of  the  compression  stresses  about  AB 
will  be 

KsXc'C. 

The  moment  of  the  tensile  stresses  about  the  same  line 
will  be 

KsXcC. 

Consequently  the  resisting  moment  of  the  whole  section 
will  bo 

M=Ks(c'C+cC)=KsXcc' (2) 

Thus  the  total  resisting  moment  is  completely  deter- 
mined. In  some  cases  of  irregular  section  the  method 
becomes  absolutely  necessary. 

It  is  to  be  observed  that  the  centre  of  gravity,  c  or  c' , 
is  at  the  same  normal  distance  from  AB  as  the  centre  of 
the  actual  stress  on  the  same  side  of  AB  with  c  or  c'. 

Art.  34. — Greatest  Stresses  at  any  Point  in  a  Beam. 

Any  beam  under  transverse  loading  is  subjected  to 
!  internal  stresses  determined  by  the  Common  Theory  of 
Flexure,  the  intensities  of  fibre  stresses  varying  directly  as 
I  the  distance  from  the  neutral  axis  while  the  transverse  and 
longitudinal  shears  are  distributed  as  indicated  in  Art.  fyo. 
The  maximum  intensities  of.  the  direct  stresses  and  shears 
at  any  point,  however,  must  be  determined  by  the  aid  of 
the  procedures  given  in  Arts.  8  and  9. 


Art.  34.]     GREATEST  STRESSES  AT  ANY  POINT  IN  A  BEAM.      163 

The  intensity  of  the  direct  tensile  and  compressive 
stresses  in  any  normal  section  may  readily  be  determined 
when  the  conditions  of  loading  are  known.  The  only 
stresses  acting  on  any  two  transverse  planes  at  right  angles 
to  each  other,  one  horizontal  and  the  other  vertical,  are 
the  direct  fibre  stress  pv  and  the  longitudinal  and  trans- 
verse shear  pxy.  It  is  shown  in  Art.  8  that  the  two  inten- 
sities of  principal  stresses  are  given  by  the  following  equa- 
tion for  all  points: 


Again,  if  a  is  the  angle  which  the  axis  of  X  (vertical) 
makes  with  the  direction  of  one  of  the  principal  stresses 
it  is  shown  in  the  same  article  that 


Pv 


(2) 


By  the  use  of  these  equations  it  is  shown  in  Art.  10 
that  at  the  neutral  surface  of  the  bent  beam  where  the 
intensity  of  the  transverse  and  longitudinal  shear  has  its 
maximum  value,  i.e.,  f  the  mean  intensity  on  the  entire 
section,  there  will  be  two  principal  stresses  of  equal  inten- 
sity, and  of  the  same  intensity  as  the  shear,  but  of  opposite 
kinds,  one  being  tension  and  one  compression,  each  making 
an  angle  of  45°  with  the  neutral  surface.  This  determines 
completely  the  state  of  stress  at  the  neutral  surface.  In 
the  same  article  it  is  shown  that  there  is  but  one  principal 
stress  at  the  exterior  surface  and  that  is  the  ordinary  fibre 
stress  of  flexure  whose  intensity  is  determined  by  the  bend- 
ing moment  at  the  normal  section  considered.  This  inten- 
sity may  be  called  k.  The  greatest  intensity  of  shearing 
stress  at  the  surface  of  the  beam  where  the  intensity  k 
exists  is  given  by  eq.  (5)  of  Art.  9.  One  of  the  principal 


164  FLEXURE.  [Ch.  II. 

stresses,  i.e.,  that  one  normal  to  the  exterior  surface  of  the 
beam  will  be  zero.  Hence  the  maximum  shear  will  be 
found  on  two  planes  at  right  angles  to  each  other  and  each 
at  45°  to  the  surface  of  the  beam,  the  intensity  of  the  shear 
being  one-half  of  the  principal  stress  k.  These  consider- 
ations determine  completely  the  greatest  stresses  at  the 
neutral  surface  and  at  the  exterior  surface,  upper  or  lower, 
of  the  beam.  There  remain  to  be  found  the  intensity  of 
principal  stress  at  all  other  points  by  means  of  eqs.  (i) 
and  (2). 

To  illustrate  the  necessary  procedures,  let  a  steel  beam 
of  rectangular  normal  section  be  taken  with  an  effective 
span  of  20  feet,  and  with  a  depth  of  16  inches.  For  the 
purpose  of  these  computations  the  beam  may  be  consid- 
ered to  have  a  lateral  thickness  or  width  of  i  inch,  making 
the  area  of  cross-section  16  square  inches.  The  load  per 
linear  foot  may  be  taken  at  1140  pounds,  producing  an 
extreme  fibre  stress  of  k  =  16,000  pounds  per  square  inch. 
If  x  be  measured  from  one  end  of  the  span  and  if  ±2  be 
measured  upward  and  downward,  respectively,  from  the 
neutral  surface,  the  greatest  value  in  either  direction  being 
8  inches,  and  if  I  be  the  moment  of  inertia  of  the  normal 
section  of  the  beam  about  its  neutral  axis,  there  may  be 
written  the  following  values  for  the  bending  moment  and 
intensity  of  fibre  stress  at  any  distance  z  from  the  neutral 
axis,  g  being  the  load  per  unit  of  span  : 


i  ' 


The  transverse  shear  at  any  section  x  from  the  end  of 
the  span  is 


Art.  34.]  GREATEST  STRESSES  AT  ANY  POINT  IN  A  BEAM.        165 

HH- 

It  is  found  by  eq.  (6)  of  Art.  15  that  the  intensity  of 
transverse  and  longitudinal  shear  at  any  point  in  a  section 
of  the  beam  is 


The  value  of  tan  2  a  giving  the  direction  in  which  the 
principal  stresses  act  now  becomes 


tan  2a=- 


.  Fig.  i  shows  a  part  of  one-half  of  the  beam  under  con- 
sideration, the  effective  span  being  20  feet  =240  inches. 
One  end  support  is  at  B  while  CD  is  at  the  centre  of  the 
span.  AW  is  a  trace  of  the  neutral  surface. 

Normal  sections  of  the  beam  were  taken  2  feet  apart 
at  F,  G,  H  and  C  and  the  directions  and  intensity  of  the 
principal  stresses  p  were  computed  by  means  of  eqs.  (i) 
and  (2)  at  four  points  2  inches  apart  vertically,  including 
the  neutral  surface  and  exterior  surfaces  at  each  of  those 
sections.  The  curved  lines  drawn  in  Fig.  i  are  each  laid 
down  in  the  direction  of  the  principal  stresses  acting  at 
each  point,  the  curves  having  the  plus  sign  representing 
the  directions  of  principal  tensile  stresses,  while  those  indi- 
cated by  the  minus  sign  show  the  directions  of  the  prin- 
cipal compressive  stresses  at  each  point.  Along  the  neutral 
surface  AW  all  lines  are  inclined  at  an  angle  of  45°  to  that 
surface,  while  at  each  exterior  surface  one  set  of  lines  is 


i66 


FLEXURE. 


[Ch.  II. 


parallel  to  that  surface  and  the  other  at  right  angles  to  it. 
Wherever  the  curved  lines  cross  they  are  at  right  angles  to 


i 


CD 


each  other.  The  plus  stresses  at  the  upper  surface  of  the 
beam  and  the  minus  stresses  at  the  lower  surface  have 
zero  intensities  at  those  surfaces.  At  the  centre  of  span 


Art.  34.]  GREATEST  STRESSES  AT  ANY  POINT  IN  A  BEAM.          167 

CD  all  lines  are  horizontal,  as  the  shear  at  that  point  is 
zero.  They  are  horizontal  whatever  may  be  the  character 
of  loading  at  the  point  where  the  bending  moment  is 
greatest,  i.e.,  where  the  shear  is  zero.  These  curved  lines 
representing  the  direction  of  the  principal  stresses  at  all 
points  are  sometimes  called  stress  trajectories. 

Some  important  practical  matters  are  based  upon  the 
existence  of  the  principal  stresses  of  tension  and  compres- 
sion at  the  neutral  surface  of  a  bent  beam,  those  principal 
stresses  making  angles  of  45°  with  that  surface.  In  Fig.  2 
is  shown  a  rolled  I-beam,  although  this  discussion  is  equally 
applicable  to  the  web  of  a  plate  girder. 

Inasmuch  as  the  inclined  principal  stresses  of  tension 
and  compression  act  at  the  neutral  surface,  let  that  dis- 
tribution of  principal  stresses  be  supposed  to  exist  through- 
out the  entire  web  of  the  rolled  beam.  This  condition  may 
be  represented  by  the  sets  of  lines  drawn  in  Fig.  2,  each  at 
an  angle  of  45°  with  the  vertical  line  (or  with  a  horizontal 
line).  Let  it  be  supposed  that  the  entire  web  of  the  beam 
is  composed  of  the  strips  shown,  those  indicated  by  the 
broken  lines  AB  being  subjected  to  tension  in  the  left  half 
of  the  bearruand  those  represented  by  full  lines,  to  compres- 
sion. Inasmuch  as  the  strips  AC  will  be  subjected  to  com- 
pression they  may  approximately  be  considered  columns 
with  the  length  h  sec  45°=/rv/2.  The  thickness  of  the 
web  of  flanged  beams  such  as  plate  girders  is  sometimes 
determined  by  an  empirical  formula  based  upon  this  long- 
column  condition  of  stress.  Any  part  AC  of  the  web  is 
in  fact  not  in  a  true  long- column  condition  because  the 
parts  parallel  to  A  B  are  in  tension  and  tend  to  hold  the 
parts  AC  in  position. 

Again,  it  is  sometimes  supposed  that  the  web  of  a  flanged 
•beam  may  be  considered  approximately  to  be  composed 
of  a  system  of  tension  and  compression  web  members  like 


1  68  FLEXURE.  [Ch.  II 

a  truss  represented  by  such  sets  of  strips  of  metal  as  AB 
and  AC. 

The  condition  of  compression  in  which  the  web  exists 
in  the  direction  AC  tends  to  buckle  a  thin  web  into  corru- 
gations with  their  axes  parallel  to  AB,  and  such  girders 
exhibit  that  result  when  tested  to  destruction  if  the  web 
is  insufficiently  stiffened.  For  this  reason  it  has  some- 
times been  proposed  to  place  the  stiffeners  on  the  webs 
of  plate  girders  in  the  direction  AC,  Fig.  2,  so  as  to 
prevent  any  buckling  of  the  kind  described.  Such  a 
method,  however,  is  not  satisfactory  for  a  number  of 
reasons. 

If  the  total  transverse  shear  in  any  normal  sections  of 
the  beam  such  as  a  vertical  section  through  A  or  C  be  called 

5  then  the  average  intensity  of  shear  assumed  uniformly 

5 

distributed  over  the  section  of  the  web  would  be  s=—  . 

th 

Since  such  a  vertical  section  would  cut  the  same  number 
of  inclined  strips  in  tension  and  compression,  the  shear 

-\/2  (sec.  45°)  would  be  carried  by  each  of  the  sets  of 
2 

inclined  strips  whose  normal  section  would  be 

th  cos  45°=-^. 


Hence,  the  intensity  of  stress  in  each  of  the  two  sets  of 
strips  would  be 

S  ,-      th       S 

_  "%/  r\    _!  __    -—    n  ___  ' 

2  '  ' 


This  is  the  same  intensity  as  the  mean  transverse  shear  on 
the  section  of  the  web.     According  to  this  mode  of  treat- 


Art.  35-1  THE  FLEXURE  OF  LONG  COLUMNS.  169 

ment,  therefore,  it  is  seen  that  the  intensities  of  stress 
throughout  the  assumed  45°  strips  is  the  same  as  the  inten- 
sity of  the  average  transverse  shear.  This  again  is  simply 
the  condition  which  exists  at  the  neutral  surface  of  the 
solid  beam  as  already  found,  except  in  that  case  the  inten- 
sity of  transverse  shear  at  the  neutral  surface  is  one  and 
one-half  times  the  average  intensity. 


Art.  35. — The  Flexure  of  Long   Columns. 

A  "long  column"  is  a  piece  of  material  whose  length 
is  a  number  of  times  its  breadth  or  width,  and  which  is 
subjected  to  a  compressive  force  exerted  in  the  direction  of 
its  length.  Such  a  piece  of  material  will  not  be  strained 
or  compressed  directly  back  into  itself,  but  will  yield 
laterally  as  a  whole,  thus  causing  flexure.  If  the  length 
of  a  long  column  is  many  times  the  width  or  breadth,  the 
failure  in  consequence  of  flexure  will  take  place  while  the 
pure  compression  is  very  small  and  neglected. 

As  with  beams,  so  with  columns,  the  ends  may  be 
"fixed,"  so  that  the  end  surfaces  do  not  change  their 
position  however  great  the  compression  or  flexure.  Such 
a  column  is  frequently,  perhaps  usually,  said  to  have 
fixed  ends.  If  the  ends  of  the  column  are  free  to  turn 
in  any  direction,  being  simply  supported,  as  flexure  takes 
place,  the  column  is  said  to  have  "round"  ends.  It  is 
clear  that  if  the  column  has  freedom  in  one  or  several 
directions  only,  it  will  be  a  "  round"  end  column  in  that 
one  direction,  or  those  several  directions,  only.  It  is 
also  evident  that  a  column  may  have  one  end  round  and 
one  end  flat  or  fixed. 

In  Fig.  i  let  there  be  represented  a  column  with  flat  ends, 
vertical  and  originally  straight.  After  external  pressure  is 


170  FLEXURE.  [Ch.  II. 

imposed  at  A,  the  column  will  take  a  shape  similar 
to  that  represented.  Consequently  the  load  P,  at 
A,  will  act  with  a  lever-arm  at  any  section  equal 
to  the  deflection  of  that  section  from  its  original 
position.  Let  y  be  the  general  value  of  that  de- 
flection, and  at  B  let  y=yr  Let  x  be  measured 
from  A,  as  an  origin,  along  the  original  axis  of 
the  column.  In  accordance  with  principles  already 
established,  the  condition  of  fixedness  at  each  of 
the  ends  A  and  C  is  secured  by  the  application  of 
a  negative  moment,  —  M.  It  is  known  from  the 
general  condition  of  the  column  that  the  curve 
of  its  axis  will  be  convex  toward  the  axis  of  x  at 
FlG'  **  and  near  A,  while  it  will  be  concave  at  and  -near 
B  (the  middle  point  of  the  column).  Hence,  since  y  is 
positive  toward  the  left,  and  since  the  ordinate  and  its 
second  derivative  must  have  the  same  sign  when  the 
curve  is  convex  toward  the  axis  of  the  abscissas,  the  general 
equation  of  moments  must  be  written  as  follows  : 

:-:•        i  ••--      -         El^M-Py  .......    (i) 

Multiplying  by  zdy, 


Py*  +  (c=o);  .     .     .     (2) 

c=o,  because  the  column  has  fiat  ends,  and 

dy_ 
dx~° 

whenj'=o.    Also 


Art.  35.]  THE  FLEXURE  OF  LONG   COLUMNS.  171 

dy 

when  y=yl ; 

•'•  M=~~ (3) 

Eq.  (2)  now  becomes 


~ 
vyy-y* 

IE  .       1 2V 

- 


•--— "         •       •      •      •     •     ^4/ 


(5, 


In  this  equation  I  is  the  length  of  the  column.  From 
eq.  (5)  there  may  be  deduced 

^EI 
r      —  p  —  .        ......      (oj 

It  is  to  be  observed  that  P  is  wholly  independent  of  the 
deflection,  i.e.,  it  remains  the  same,  whatever  may  be  the 
amount  of  deflection,  after  the  column  begins  to  bend. 
Consequently,  if  the  elasticity  of  the  material  were  per- 
fect, the  weight  P  would  hold  the  column  in  any  posi- 
tion in  which  it  might  be  placed  after  bending  begins. 
This  result  is  for  pure  flexure,  direct  compression  being 
neglected. 

Eq.  (6)  forms  the  basis  of  some  old  long  column 
formulae  now  out  of  use.  It  was  first  established  by 
Euler. 


i?2  FLEXURE.  [Ch.  II. 

Some  very  important  results  follow  from  the  conside- 
ration of  Fig.  i  in  connection  with  the  preceding  equa- 
tions. 

The  bending  moment  at  the  centre,  B,  of  the  column 
is  obtained  by  placing  y=yl  in  eq.  (i);  its  value  is,  con- 
sequently, 

M  ......     (7) 


Hence  the  bending  at  the  centre  of  the  column  is  exactly 
the  same  (but  of  opposite  sign)  as  that  at  either  end.  Between 
A  and  B,  then,  there  must  be  a  point  of  contra-flexure. 

Putting  the  second  member  of  eq.  (i)  equal  to  zero, 
and  introducing  the  value  of  M  from  eq.  (3), 


Introducing  this  value  of  y  in  eq.  (4),  and  bearing  in 
mind  eq.  (5), 


TT   \~E~I     I 
-V- 


The  points  of  contra-flexure,  then,  are  at  H  and  D, 
JZand  |/  from  A. 

Hence  the  middle  half  of  the  column  (HD)  is  actually  a 
column  with  round  ends,  and.  it  is  equal  in  resistance  to  a 
fixed-end  column  of  double  its  length. 

Hence  writing  /'  for  -  and  putting  2/'  for  /  in  eq.  (6), 


(9) 


Eq.  (9)  gives  the  value  of  P  for  a  round-end  column. 
Again,   either  the  upper  three   quarters    (AD)    or  the 
lower  three  quarters  (CH)  of  the   column   is  very  nearly 


Art.  35.]  THE    FLEXURE   OF   LONG   COLUMNS.  173 

equivalent  to  a  column  with  one  end  flat  and  one  end  round, 
and  its  resistance  is  equal  to  that  of  a  fixed-end  column 

whose  length  is  -  its  own.     Putting,  therefore, 


and  introducing 

in  eq.  (6), 

n*EI 


The  last  case  is  not  quite  accurate,  because  the  ends  of 
the  columns  HC  and  AD  are  not  exactly  in  a  vertical  line. 

In  reality,  the  column  under  compression  may  be  com- 
posed of  any  number  of  such  parts  as  HD,  with  the  por- 
tions HA  and  CD  at  the  ends,  thus  taking  a  serpentine 
shape,  so  far  as  pure  equilibrium  is  concerned.  In  such 
a  condition  the  column  would  be  subjected  to  considerably 
less  bending  than  in  that  shown  in  the  figure.  In  ordinary 
experience,  however,  the  serpentine  shape  is  impossible, 
because  the  slightest  jar  or  tremor  would  cause  the  column 
to  take  the  shape  shown  in  Fig.  i.  Hence  the  latter  case 
only  has  been  considered. 

If  r  is  the  radius  of  gyration  and  5  the  area  of  normal 
section  of  the  column,  eqs.  (6)  and  (9)  will  take  the  forms 

P     47T2£r2  P  _7r2£r2 

"5  ~  ~P~  "5  =  ~W' 

Eq.  (10)  will,  of  course,  take  a  corresponding  form. 

P 
These  equations  evidently  become  inapplicable  when  ~- 


174  FLEXURE.  [Ch.  II. 

approaches  C,  the  ultimate  compressive  resistance  of  the 
material  in  short  blocks.  The  corresponding  values  of  f-j 
at  the  limit  are 

M!    -    •    ;    (". 


for  fixed  and  round  ends  respectively ;   other  conditions  of 
ends  will  be  included  between  those  two. 
If  for  structural  steel 

£=30,000,000     and     C  =  6o,ooo, 

the  above  values  become  140  and  70,  nearly. 

Euler's  formula,  therefore,  is  strictly  applicable  only  to 
structural  steel  columns,  with  ends  fixed  or  rounded,  fox 
which  l  +  r  greatly  exceeds  140  and  70,  respectively. 

If  for  cast  iron 

£  =  14,000,000     and     C  =  100,000, 
eqs.  (n)  give 

I  I 

-  =  74     and     -=37,  nearly. 

Euler's  formula  evidently  becomes  inapplicable  con- 
siderably above  the  limits  indicated,  since  columns  in  which 

—  has  those  values  will  not  nearly  sustain  the  intensity  C. 

The  analytical  basis  of  "  Gordon's  Formula"  for  the 
resistance  of  long  columns  is  so  closely  associated  with  the 
empirical  that  both  will  be  treated  together  hereafter. 


Art.  36.]     SPECIAL  CASES  OF  FLEXURE  OF  LONG  COLUMNS.         175 


Art.  36. — Special  Cases  of  Flexure  of  Long  Columns. 

There  are  a  few  cases  of  flexure  of  columns  which, 
while  not  frequently  found  in  engineering  experience,  may 
be  of  some  practical  importance.  The  two  or  three  which 
follow  involve  the  integration  of  linear  differential  equations 
treated  in  advanced  works  on  the  integral  calculus;  con- 
sequently the  operations  of  integration  will  not  be  given 
here,  but  the  general  integrals  will  be  assumed. 

Flexure  by  Oblique  Forces. 

In  Fig.  i  let  OA  represent  a  column  acted  upon  by  the 
oblique  force  P,  which  makes  the  angle  a  with  the  axis  of 
X.  The  column  is  supposed  to  be 
fixed  in  the  direction  of  OX  at  0,  but 
the  coordinates  oo  and  y  are  measured 
from  the  point  of  application  A  of  the 
load  P  as  shown  in  the  figure.  If 
right-hand  moments  are  positive,  and 
left-hand  negative,  the  component 
P  sin  a  will  have  the  negative  moment 

—  P  sin  ax  about  any  point  0' '.     The 
lever  arm  of  P  cos  a,  if  the  deflection 
y  is  positive,  is  -\-y,  and  its  moment 

—  P  cos  ay  is  also  negative.       Hence 
the  resultant  moment  of  any  force,  P, 
in  reference  to  the  point  0'  is 


ax2 

00  — P  COS  ay 


d) 


FIG.  i. 


For  any  number  of  forces  or  loads  P  there  will  obviously 
be  a  corresponding  number  of  pairs  of  terms  in  the  third 


i76 


FLEXURE. 


[Ch.  II. 


member  of  eq.  (i).     It  will  therefore  be  sufficient  to  treat 
one  force  P  only. 

Eq.  (i)  may  be  put  in  the  form, 

P  sin  «  /  x 


0 
In  this  equation  n2  = 


El 

P  COS  a 


-,  P  sin  a       ^        /  N 

and  m=  .     Eq.  (2) 


may  readily   be  integrated   so   as   to   give   the   following 
equation,  Ci  and  €2  being  constants  of  integration  : 


m 
n3 


sn  noc  —    z  cos  n% 


•     (3) 


Using  the  values  of  m  and  n  given  above  y  may  take 
the  following  form,  observing  that  —  =  —tan  a: 


The  coefficients  C  and  C'  have  the  values 

~EI 


and 


and  they   may  be  treated   as   arbitrary  constants  to  be 
determined  by  the  conditions  of  each  problem 

As  x  and  the  deflection  y  are  measured  from  the  point 
of  application  of  the  load  P,  if  x  =o  then  must  y=o.  Hence 
by  eq.  (4),  C'=o.  Consequently 


Art.  36.]     SPECIAL  CASES  OF  FLEXURE   OF  LONG  COLUMNS.        177 

If  a  is  greater  than  90°,  cos  a  will    be  negative   and 
the  exponential  value  of  the  sine  may  be  used  as  follows: 

Placing  b  =  */ — cT"""'  anc^  e  being  the  base  of  the  Naperian 

\       tLL 

logarithms : 

(6) 


2         —  I 

When  cos  «  is  negative  bV  —  i  is  the  square  root  of  a 

positive  quantity,  and      -  will  be  rational. 

V  —i 

Column  Free  at  Upper  End  and  Fixed  Vertically  at  Lower 
End  with  either  Inclined  or  Vertical  Loading  at  Upper  End. 

In  this  case  the  axis  of  x,  Fig.  i,  is  to  be  considered 
vertical  with  the  column  fixed  at  its  base  0.     In  accord- 

ance with  the  latter  condition  -~-  =o  at  0,  i.e.,  when  x  -I  = 

ax 

length  of  column. 
From  eq,  (5), 


(7) 
dx     " v     El         '  \     El 

It  is  to  be  observed  that  P  is  not  yet  determined  and 

/  p 
that  cos\f — ^-^  x  may  vary  largely  (and    periodically) 


while  -f-  remains  unchanged. 
doc 

If  the  column  carries  a  vertical  load  at  its  upper  end 
a  =  o  =  tan  a,  and  when  x  =  l,-¥=o.     Eq.  (7)  then  gives  : 


Cofi%/i= 


178  FLEXURE.  [Ch.  II. 

If  /  is  any  whole  odd  number  from  i  to  infinity,  then 
there  may  be  placed  by  the  aid  of  eq.  (8) : 

~P         fir 

J  (      \ 

EI  =  7l :     .     .     .     (9) 


If  this  value  be  substituted  in  eq.    (7)   after  making 
a  =  tan  a  =  o : 


Eq.  (10)  shows  that  when  #=— (/  being  any  whole  odd 

number)  -r  =o,  for  cos  -  =cos  90°  =o. 
dx  2 

Obviously  P  must  have  the  smallest  value  which  will 
satisfy  eq.  (9) ;  but  /  cannot  be  smaller  than  i.     Therefore 

P=EI^.  riii 


The  carrying  capacity  of  the  column  is  thus  seen  to  be 
independent  of  the  deflection  as  was  the  case  in  Art.  35, 
but  it  must  be  observed  that  the  effect  of  direct  compression 
is  neglected,  i.e.,  it  is  a  case  of  pure  bending  of  excessively 
long  columns.  The  end  of  the  column  considered  here 
which  carries  the  vertical  load  is  free  to  deflect  laterally, 
whereas  in  Art.  35  both  ends  are  supposed  to  be  held 
against  lateral  movement.  In  the  latter  case  the  resist- 
ance is  seen  to  be  nine  times  as  great  as  in  the  present. 

Eq.  (n)  can  be  found  in  a  direct  and  simple  manner 
by  making  M  =o  in  eq.  (i)  of  Art.  34  and  integrating  the 
resulting  equation. 


/  P  /~P~ 

Since  by  eq.  (8),  cos^|— /=o,  sin^— 


Art.  36.]    SPECIAL  CASES  OF  FLEXURE  OF  LONG    COLUMNS.         179 

If  therefore  a=o  and  x=l  in  eq.  (5),  and  if  y  is  the 
deflection  of  the  free  end  of  the  column  in  reference  to 
the  base,  Fig.  i,  that  equation  will  give: 

C=yi (12) 

Then 


(13) 


For  a  given  value  of  x,  therefore,  y  varies  directly  as 
yi  and  the  relative  deflections  at  the  base  and  any  point 
may  be  computed  by  the  equation: 

£-W£A  (i4) 


Or  in  the  ordinary  case: 

J-rinJ'.  — ;.'.    .     -     -     (IS) 

It  should  be  remembered  that  deflection  is  initiated  by 
the  load  P  determined  by  eq.  (n)  and  that  the  deflection 
may  take  any  subsequent  value  without  increase  of  load. 

PROBLEMS  FOR  CHAPTER  II. 

Problem  i. — A  beam  simply  supported  at  each  end 
carries  a  load  of  850  pounds  per  linear  foot  over  a  span  of 
26  feet.  Find  the  bending  moment  and  transverse  shears 
at  the  end  and  centre  of  span  and  at  2  points  3  feet  and  1 1 
feet  6  inches  respectively  from  the  end. 

Ans.  Moment  at  end  is  o;  at  3  feet,  29,325  ft.-lbs.; 
at  11.5  feet,  70,868.75  ft.-lbs.;  at  centre,  71,825 
ft.-lbs.  Shear  at  end  is  11,050  Ibs.;  at  3  feet, 
8500  Ibs.;  at  11.5  feet,  1275  Ibs. 


i8o  FLEXURE.  [Ch.  II. 

Problem  2. — A  beam  or  girder  having  a  span  length  of 
41  feet  carries  a  uniform  load  of  1200  pounds  per  linear  foot 
and  a  single  weight  of  1800  pounds  at  the  centre.  Find 
the  bending  moments  and  the  shears  due  to  the  uniform 
load  and  the  single  load  separately  at  the  ends  and  at  the 
centre  and  at  points  6  and  14  feet  from  the  end. 

Problem  3. — In  Problem  2  find  the  single  weight  which 
placed  at  the  centre  of  the  span  will  produce  the  same 
centre  bending  moment  as  the  uniform  load. 

Ans.  24,600  pounds. 

Again,  find  two  weights  placed  6  feet  apart,  i.e.,  one 
3  feet  either  way  from  the  centre,  which  will  produce  the 
same  centre  bending  moment  as  the  uniform  load. 

Ans.  Each  of  the  two  weights  is  14,406  pounds. 

Problem  4. — A  beam  or  girder  with  a  span  length  of 
31  feet  carries  a  uniform  load  of  300  pounds  per  linear  foot 
in  addition  to  five  loads,  the  first  weighing  7000  pounds 
at  a  distance  of  3  feet  from  the  end ;  the  second  weighing 
10,000  pounds  7  feet  from  the  end;  the  third  weighing 
11,000  pounds  14  feet  from  the  end;  the  fourth  weighing 
17,000  pounds  21  feet  from  the  end,  and  the  fifth  weighing 
6400  pounds  27  feet  from  the  end. 

Construct  the  shear  and  moment  diagrams  for  this  case, 
Fig.  2  of  Art.  15  and  Fig.  2  of  Art.  12. 

Problem  5. — Find  a  uniform  load  for  the  same  beam 
considered  in  Problem  4  which  will  have  a  centre  bending 
moment  equal  to  the  greatest  bending  moment  of  that 
problem;  also  another  uniform  load  whose  end  shear 
shall  equal  the  greatest  of  the  two  end  shears  of  Prob- 
lem 4.  Such  uniform  loads  are  called  "  equivalent  uniform 
loads." 

Problem  6. — In    Problem   2  the  moment  of  inertia  / 


Art.  36.)     SPECIAL    CASES   OF  FLEXURE   OF  LONG   COLUMNS.     181 

is '3570  (the  unit  being  the  inch),  while  E  =  30,000,000, 
the  beam  being  of  steel.  Find  the  tangent  of  inclination 
of  the  neutral  surface  at  the  end  and  at  10  feet  from  the 
end.  Also  find  the  deflection  at  the  centre  of  span  and 
at  10  feet  from  the  end.  Use  eqs.  (19),  (20),  and  (21) 
of  Art.  22. 

Partial  Ans.  Tangent  10  feet  from  end  is  .00344. 
The  deflection  at  the  same  point  is  .53  inch. 

Problem  7. — In  Problem  6  let  it  be  required  to  ascer- 
tain how  much  additional  deflection  is  produced  by  the 
transverse  shear  at  the  centre  of  the  span  and  at  10  feet 
from  the  end.  Let  the  coefficient  of  elasticity  for  shear 
(G)  be.  taken  at  12,000,000  pounds,  while  /  =  357o  and 
d  =  i4  inches. 

Ans.  Deflection  at  10  feet  is  .0054  inch,  and  at  the 
centre  of  span  .0075  inch. 


CHAPTER  III. 
TORSION. 

Art.  37. — Torsion  in  Equilibrium. 

THE  state  of  stress  called  torsion  is  produced  when  a 
straight  bar  of  material,  like  a  piece  of  round  shafting,  is 
twisted.  Such  a  bar  is  represented  in  Fig.  i,  the  axis  of 
the  piece  being  AB,  and  its  normal  cross-section  having 
any  shape  whatever.  In  engineering  practice  the  outline 
of  that  normal  section  is  usually  circular,  although  it  is 
occasionally  square. 


FIG.   i. 

The  twisting  of  the  bar  is  done  by  the  action  of  two 
equal  and  opposite  couples  acting  in  two  planes,  each  nor- 
mal to  the  axis,  but  at  any  desired  distance  apart.  The 
two  couples  are  represented  in  Fig.  i  at  each  end  of  the 
piece  in  the  two  normal  sections  A  and  B.  The  forces 
and  lever-arm  of  one  couple  are  respectively  P  and  £,  and 
Pf  and  er  of  the  other.  The  moment  of  the  first  couple 

182 


O       <4 


bfl  o        $ 
'*'%      £ 

.M  -n     ° 
rt  £    ^ 

o 

CH      rt 
^      *-" 


86 


•-'  -S 
J  -2 
^7, 


^   M 
TJ 

0 


"3   -2 

^  _  , 


111 


Art.  37-J  TORSION  IN  EQUILIBRIUM.  183 

will  be  Pe  and  that  of  the  second  couple  PV,  and  if  pure 
torsion  is  to  be  produced  these  two  moments  must  be  equal, 
but  opposite  to  each  other.  Inasmuch  as  the  moment  of  a 
couple  is  the  product  of  the  force  by  the  lever-arm,  the 
forces  and  lever-arms  of  the  two  twisting  couples  may  vary 
to  any  extent  as  long  as  the  moments  remain  unchanged. 

Although  the  system  of  forces  to  which  a  bar  in  torsion 
is  subjected  is  such  as  to  be  in  equilibrium,  any  portion 
of  the  piece  will  tend  to  have  its  normal  sections  like  those 
at  CD  rotated  over  each  other,  the  result  being  a  small 
sliding  motion  around  the  axis  of  the  piece.  Hence  a 
torsive  stress  is  wholly  a  shearing  stress  on  normal  sections 
of  the  piece  subjected  to  torsion.  It  is  further  important 
to  observe  that  inasmuch  as  a  couple  produces  the  same 
effect  wherever  it  may  act  in  its  own  plane,  the  actual 
twisting  moment  need  not  be  applied  with  its  forces  sym- 
metrically disposed  in  reference  to  the  axis  of  the  piece; 
indeed,  both  of  those  forces  may  be  anywhere  on  one  side 
of  the  piece  without  varying  the  conditions  of  torsion  or 
torsive  stress  to  any  extent  whatever. 

It  is  known  from  the  general  theory  of  stress  in  a  solid 
body  that  although  there  can  be  no  stresses  of  tension  and 
compression  parallel  to  the  axis  of  a  bar  under  torsion,  or 
at  right  angles  to  it,  there  will  be  such  stresses  of  varying 
intensities  on  oblique  planes.  Inasmuch  as  the  result  of 
torsion  is  to  slide  normal  sections  each  past  its  neighbor, 
the  elastic  torsive  shear  like  any  other  shear  will  not  change 
the  volume  of  the  body.  The  principal  shearing  strains 
will  produce  deformation  without  changing  the  dimensions 
whose  product  gives  the  volume. 

The  exact  and  complete  mathematical  theory  of  tor- 
sion deduced  from  the  general  equations  of  equilibrium  of 
stresses  in  an  elastic  solid,  without  extraneous  assump- 
tions, will  be  found  in  App.  I.  Those  formulas  show  accu- 


'84  TORSION.  [Ch.  III. 

rately  the  state  of  torsive  stress  in  bars  of  any  elastic 
material  and  of  various  shapes  of  cross-section.  For  the 
general  purposes  of  engineering  practice  that  general  demon- 
stration is  rather  complicated.  Hence  it  is  often  avoided 
by  making  certain  approximate  assumptions  based  to.  some 
extent  on  experimental  observations  which  lead  to  an 
approximate  and  simpler  theory,  yielding  formulae  accurate 
only  for  the  circular  normal  section,  but  which  are  not 
materially  in  error  for  the  square  section.  These  formulas 
are,  however,  far  from  accurate  for  certain  other  sections. 
In  this  article  only  the  formulas  of  the  simpler,  theory, 
called  the  common  theory  of  torsion, 
will  be  given. 

Fig.  2  is  supposed  to  represent  the 
normal  section  of  a  bar  of  material  of 
any  shape,  subjected  to  torsion  by  the 
application  of  couples  as  shown  in 
Fig.  i.  The  fundamental  assump- 
tions of  the  common  theory  of  tor-  FIG.  2. 
sion  are  that  the  intensity  of  shearing  stress  varies  directly  as 
the  distance  from  a  central  point  at  which  that  intensity  is 
zero,  and  that  that  central  point  is  located  at  the  centre  of 
gravity  or  the  centroid  of  the  section.  It  is  also  implicitly 
assumed  that  the  normal  sections  which  are  plane  before 
torsion  remain  plane  during  torsion.  In  Fig.  2,  A  is  sup- 
posed to  be  the  centre  of  gravity  of  the  section  at  which  the 
intensity  of  shear,  i.e.,  the  shear  per  square  unit  of  section, 
is  zero.  The  distance  from  the  centre  A  to  any  point  of 
the  section  is  represented  by  r,  and  to  the  most  remote 
point  in  the  perimeter  of  the  section  by  r0.  In  accord- 
ance with  the  assumed  law,  the  greatest  intensity  of  shear 
T m  in  the  section  will  be  found  at  the  distance  rQ  from  its 
centre.  While  this  is  accurately  true  for  the  circular  sec- 
tion, it  is  quite  erroneous  for  a  number  of  other  sections. 


Art.  37.1  TORSION  IN  EQUILIBRIUM.  185 

Hence  the  intensity  at  the  distance  unity  from  the  centre 
A  will  be  -  — ,  and  at  the   distance    r  from  the   centre  it 

ro 

will  have  the  value 


The  element  of  the  section  at  the  distance  r  from  A  will  be 

rdco.dr  ........     (2) 

Hence  the  shear  on  that  element  is 

T"* 

dS=~Tm.rdw.dr  =  —  r*dr.da>.     ...     (3) 
ro  ro 

The  direction  of  action  of  this  torsive  shear  is  around 
the  circumference  of  a  "circle  whose  radius  is  r;  hence  if 
moments  of  all  these  small  shears,  dS,  be  taken  about  the 
centre  or  point  of  no  shear,  A,  the  lever-arm  of  each  small 
force,  dS,  will  be  r,  and  the  differential  moment  will  be 

dM=rdS  =  —r3dr.daj.       .     .     .     .     (4) 

ro 

The  total  moment  of  torsion  therefore  will  be 


/27r      rr     -T  T         /»2»r     /V0  7- 

/      —rtdr.dto^^  /     r*drdco=^I,.     (5) 

Jo       rQ  rQ  Jo     Jo  r0   *     VD/ 

The  quantity  Ip  is  the  polar  moment  of  inertia  of  the  section. 
For  a  circular  section 

7rr04     nd4  Ad2 

lp=-=—(d  =  diameter)  =--.       .     .     (6) 


1  86  TORSION.  [Ch.  111. 

For  a  square  section  (b  =  side  of  square) 


Ab 


For  a  rectangular  section  (6=  one  side  and  c  =  the  other 
side) 

' 


....     (8) 


J2  I2 

For  an  elliptical  section  (a,  and  bl  being  semi-axes) 


(9) 


Using  the  notation  of  Fig.  i,  the  following  equation  of 
moments  may  be  written,  Pe  being  the  moment  of  the 
external  twisting  couple  and  M  the  moment  of  the  internal 
torsive  shearing  stresses  in  any  normal  section: 


(10) 


It  is  clear  from  Art.  2,  if  <p0  is  the  shearing  strain  at  the 
distance  r0  from  the  centre,  that  Tm=G</>0,  G  being  the 
coefficient  of  elasticity  for  shearing.  Also,  since  the  inten- 
sity of  shearing  varies  directly  as  the  distance  from  the 
centre  A,  it  is  equally  clear  that  the  shearing  strain  </> 
varies  directly  as  the  distance  from  the  centre,  so  that 
if  a  represents  the  shearing  strain  at  unit's  distance  from  A 

(j)=ra     and*   </>Q=rQa  .....     (n) 

Hence  in  general 

T=Gra,    .......     (12) 

and  as  a  maximum 

Tm=Gr0a  .......     (13) 


Art.  37.]  TORSION  IN  EQUILIBRIUM.  187 

a  is  evidently  the  angle  through  which  one  end  of  a  fibre 
of  unit  's  length  and  at  unit  's  distance  from  the  centre  or  axis 
is  turned.  It  is  called  the  angle*  of  torsion. 

If  /  is  the  length  of  the  piece  twisted,  the  total  angle 
through  which  the  end  of  the  fibre  at  unit's  distance  from 
the  axis  will  be  turned  is 

Total  angle  of  torsion  =a<,.       .     .     .     (14) 

If  the  fibre  is  at  the  distance  r0  from  the  axis  one  end 
willbe  twisted  around  beyond  the  other  by  an  amount 
equal  to 

Total  strain  of  torsion  =  rQal.    .     .     .     (15) 

By  the  aid  of  eq.  (13)  eq.  (5)  may  be  written 

I,.       .     .     .  '    .     (16) 

o 

If  <£0  is  observed  experimentally 

*« 


The  angle  through  which  a  shaft  will  be  twisted  by  the 
moment  Pe,  if  the  length  is  /,  is 

Pel      lTm 


If  G  is  in  pounds  per  square  inch,  as  is  usual,  the  pre- 
ceding formulae  require  all  dimensions  to  be  in  inches, 
while  a  will  be  arc  distance  at  radius  of  one  inch. 

If  i2l  is  written  for  /  the  unit  for  the  latter  quantity 
must  be  the  foot. 

By  inserting  the  value  of  1P  from  eq.  (6)  in  eq.  (5), 

*  This  small  angle  is  measured  in  radians.     Strictly  speaking  it  is  an 
indefinitely  short  arc  with  unit  radians, 


1  88  TORSION.  [Ch.  III. 


. 

d      32          16 

? 

~.     ......     (19) 

w 

Eq.  (19)  will  give  the  diameter  of  a  shaft  capable  of 
resisting  the  twisting  moment  represented  by  Pe  with  the 
maximum  torsive  shear  in  the  extreme  fibres  of  Tm. 

The  main  cross  dimensions  of  other  sections  may  be 
found  similarly  by  the  use  of  eqs.  (7),  (8),  and  (9). 

It  is  frequently  convenient  to  compute  the  greatest 
intensity  Tm  from  the  twisting  moment  M.  For  this  pur- 
pose the  equation  preceding  eq.  (19)  gives 

M 

^m  =  5-1^?  .......       (20) 

These  equations  complete  all  that  are  required  for  the 
practical  use  of  the  common  theory  of  torsion.  In  some 
cases  it  may  be  necessary  to  use  accurate  formulae  for 
other  shapes  of  section  than  the  circular.  In  those  cases 
the  exact  formulas  of  App.  I  should  be  employed.  The 
practical  applications  of  the  preceding  formulae  to  such 

Twisting  Moment  in  Terms  of  Horse-power  H. 

It  is  sometimes  convenient  to  express  the  twisting 
moment  M  in  terms  of  horse-power  transmitted  by  the 
snafting.  If  H  is  the  number  of  horse-powers  transmitted 
by  a  shaft  making  n  revolutions  per  minute,  the  inch-pounds 
of  work  will  be  12  X33,oooX#,  since  each  horse-power  repre- 
sents 33,000  foot-pounds  of  work  performed  per  minute. 
Again  if  e  is  the  lever  arm  of  the  twisting  couple,  the  path 
of  the  force  P  per  minute  will  be  2  wen  and  the  work  per- 
formed by  the  couple  must  therefore  be  PX2Tren=M2wn. 
Equating  these  two  expressions  for  the  work  or  energy 
transmitted  ; 


Art.  37-]  TORSION  IN  EQUILIBRIUM.  189 

,  H      ,,  ,     x 

=63,025—  =M.    .     .     .     (21) 


2-n-n  n 

If  this  value  of  M  be  placed  in  eqs.  (19)  and  (20),  the 
values  of  d,  the  diameter  of  the  shaft  and  Tm,  the  greatest 
intensity  of  shear  will  take  the  following  forms  in  terms 
of  the  horse-power  and  the  number  of  revolutions  per 
minute  : 

31 

-.    .     .....     (22) 

n 

TT 

••      (23) 


Hollow  Circular  Cylinders.    • 

If  the  exterior  diameter  of  a  hollow  cylinder  is  d  and  the 
interior  diameter  d\  =jd,  j  being  simply  the  ratio  between 
the  two  diameters,  the  equation  preceding  eq.  (19)  may  be 
written : 

M  =  — ^(d3  —  di3) (24) 

Hence 

Eq.  (25)  shows  that  any  of  the  preceding  equations 
may  be  made  applicable  to  a  hollow  cylinder  by  writing 
Tm(i  -j3)  in  the  place  of  7W. 

Eqs.  (19)  and  (22)  therefore  take  the  following  forms 
for  a  hollow  cylinder : 


Pe  ,       a         H  ,  <.\ 

•   •    (26) 


The  resistance  of  the  hollow  cylinder  is  obviously  the 
difference  between  the  resistances  of  two  solid  cylinders, 


19°  TORSION,  [Ch.  III. 

one  having  the  exterior  diameter  and  the  other  the  interior 
diameter  of  the  hollow  cylinder.    * 


Art.  38. — Practical  Applications  of  Formulae  for  Torsion. 

There  has  been  comparatively  little  experimental  inves- 
tigation in  the  resistance  of  structural  materials  to  torsion 
and  practically  none  of  that  has  been  done  in  connection 
with  pieces  of  considerable  size.  Such  results  as  have  been 
obtained  appear  to  justify  the  following  data. 

Steel. 

Some  of  the  older  tests,  as  those  of  Kirkaldy,  indicate 
that  the  ultimate  intensity  of  torsional  shear,  Tm,  may  be 
taken  as  high  as  75,000  pounds  to  90,000  pounds  per 
square  inch  for  special  grades  of  steel  like  those  used  for 
tires,  rails,  and  crucible  steel,  but  lower  values  must  be 
employed  for  mild  structural  steel  and  for  the  ordinary 
grades  of  shafting. 

Torsion  tests  on  circular  pieces  of  spring  and  cold-drawn 
steel  about  f  inch  and  ij  inches  in  diameter  made  in  the 
testing  laboratory  of  the  Dept.  of  Civil  Engineering  at 
Columbia  University  by  Mr.  J.  S.  Macgregor  gave  the 
following  results,  which  are  shown  rather  fully  in  order  to 
exhibit  clearly  their  main  features.  There  were  either  four 
or  six  tests  in  each  group  from  which  the  "  max.,"  "  mean  " 
and  "-min."  were  taken.  All  these  test  specimens  except 
those  of  mild  steel  were  heat  treated.  Part  of  these  were 
heated  to  1350°  F.  and  then  plunged  in  oil  at  70°  until  cold. 
They  were  then  temper  drawn  in  hot  oil  at  575°  F.  and 
part  were  again  heated  to  1350°  F.  and  immersed  in  oil  at 
575°  F,  They  were  then  allowed  to  cool  in  air  at  normal 
temperature. 


Art.  38.]        APPLICATIONS  OF  FORMULA  FOR  TORSION. 


191 


Diam. 
Inches. 

Pe. 
In.-Lbs. 

Tm- 
Elastic. 

Tm- 

Ult. 

Modulus. 
G. 

(max. 
mean 

.617 

5,640 

5,427 

41,600 
4O,7IO 

122,310 

118,110 

13,010,000 
12,455,000 

min. 

.614 

5,290 

31,050 

114,620 

11,292,000 

(max. 
mean 

1.252 

45,260 
AA.OAO 

46,100 
43,I3O 

117,320 
115,070 

13,954,500 

.'12,659  ooo 

min. 

I  .246 

42,720 

41,500 

111,900 

11,830,000 

(  max. 
Cold-drawn  steel.   \  mean 
[  min. 

1.252 
1-25 

43,500 
37,990 
34,270 

46,200 
39,300 

33,ooo 

113,200 
99,000 
89,500 

12,445,000 
11,602,000 
10,534,000 

f  max. 
Mild  steel                  -1  mean 

1-257 

24,500 
23,200 

22,000 
20,800 

62,800 
60,050 

12,600,000 

12,  IIO  OOO 

[  min. 

1-233 

21,520 

I9,7oo 

57,300 

11,700,000 

It  has   been  shown  in  Art.   5  that    r  =  —  —  i.      Hence 

2Cr 

if  £=30,000,000,  which  is  essentially  correct  for  steel,  and 
if  £  =  12,000,000  as  the  mean,  approximately,  of  the  values 
in  the  preceding  table,  then  will 


Direct  torsion  tests  of  six  small  nickel  steel  specimens 
by  Prof.  E.  L.  Hancock  and  described  by  him  in  Vol.  VI 
(1906)  of  Proceedings  of  the  American  Society  for  Testing 
Materials  gave  elastic  limits  : 

Max.  Mean.  Min. 

Nickel  Steel  .  .36,000      32,900      30,500  pounds  per  sq.in. 

He  also  found  for  mild  carbon  steel  the  two  following 
elastic  limits: 

Mild  Carbon  Steel.  .  .  .  29,000.  .  .  .25,500  pounds  per  sq.in. 

As  the  ultimate  resistance  of  mild  carbon  steel  to  torsi  ve 
or  ordinary  shear  may  be  taken  at  about  three-quarters 


192  TORSION.  [Ch.  Ill 

the  ultimate  tensile  resistance,  and  approximately  the  same 
ratio  between  the  elastic  limits,  it  is  reasonable  to  take  the 
elastic  limit  in  torsion  at  25,000  pounds  to  28,000  pounds 
per  square  inch  for  that  grade  of  material  having  an  ulti- 
mate tensile  resistance  of  60,000  pounds  to  68,000  pounds 
per  square  inch. 

Nickel  steel  has  a  higher  ratio  of  the  elastic  limit  divided 
by  the  ultimate,  and  a  mean  value  of  33,000  pounds  per 
square  inch  for  the  elastic  limit  is  reasonable. 

If  the  greatest  intensity  of  torsive  shear  Tm  allowed  in 
the  design  of  a  shaft  of  diameter  d  is  fTe  in  which  Te  is 
the  elastic  limit  and/  a  suitable  fraction,  perhaps  .5  in  some 
cases,  then  eq.  (19)  of  the  preceding  article  will  take  the 
form: 

Pe 


Similarly  eq.  (22)  of  the  same  Art.  will  become: 
3~fi 


Wrought  Iron. 

Wrought  iron  is  now  seldom  used  for  shafting  or  similar 
purposes,  but  such  tests  as  have  been  made  show  that  the 
torsive  elastic  limit  of  wrought  iron  may  be  taken  from 
20,000  pounds  to  25,000  pounds  per  square  inch  and  used 
as  indicated  in  eqs.  (i)  and  (2).  From  10  per  cent  -to  20 
per  cent  higher  values  may  be  taken  for  cold-rolled  shafting. 

Cast  Iron. 

Cast  iron  is  ill  adapted  to  resist  torsion  and  is  not 
commonly  used  for  that  purpose,  yet  it  has  been  tested 


Art.  38.]        APPLICATIONS  OF  FORMULA  FOR   TORSION.  193 

in  torsion,  although  generally  in  special  grades  such  as  were 
formerly  employed  in  making  cannon  or  car  wheels.  Such 
grades  of  cast  iron  gave  ultimate  values  of  Tm  from  24,000 
pounds  to  45,000  pounds  per  square  inch  or  even  more, 
but  they  are  far  too  high  for  ordinary  castings  used  in 
engineering  practice.  Probably  half  the  preceding  values 
would  be  large  enough  for  the  best  quality  of  ordinary 
castings,  although  the  highly  variable  and  erratic  qualities 
of  cast  iron  make  it  exceedingly  difficult  to  assign  exact 
data  for  purposes  of  design.  The  modulus  of  elasticity,  G, 
may  be  taken  at  7,000,000  for  ordinary  grades  of  cast  iron, 
or  at  6,000,000  for  the  lower  grades. 

Alloys  of  Copper,  Tin,  Zinc  and  Aluminum. 

The  torsional  resistance  of  this  class  of  alloys  varies 
greatly  with  the  relative  proportions  of  their  constituent 
elements  in  a  manner  quite  similar  to  that  exhibited  by 
the  corresponding  resistance  to  tension. 

Professor  R.  H.  Thurston  was  probably  the  earliest 
thorough  investigator  of  the  torsional  resistances  of  many 
of  these  alloys.  He  found  the  ultimate  intensity  of  torsi ve 
stress  Tm  to  vary  from  a  few  hundred  pounds  per  square 
inch  to  nearly  48,000  pounds  per  square  inch  for  alloys  of 
copper  and  tin  running  by  gradual  variation  from  pure  cop- 
per to  10%  of  that  metal  alloy  to  90%  of  tin.  The  alloy 
80-90%  Cu  with  20-10%  vSn  gave  Tm  varying  from  about 
47,700  pounds  to  43,900  pounds  per  square  inch  with  a  maxi- 
mum twist  of  114.5  degrees.  Similarly  he  found  the  ulti- 
mate Tm  for  pure  copper  to  range  from  28,400  to  35,900 
pounds  per  square  inch  with  a  total  twist  of  over  150  de- 
grees. On  the  other  hand,  pure  tin  gave  the  ultimate 
TTO=32oo  pounds  (nearly),  the  total  angle  of  twist  running 
as  high  as  691  degrees.  The  elastic  limit  of  the  more  due- 


194 


TORSION. 


[Ch.  III. 


tile  of  these  alloys  was  found,  to  vary  from  about  35%  to 
60%  of  the  ultimate  Tm.  The  alloys  running  from  70% 
Cu  with  30%  Sn  to  29%  Cu  with  71%  Sn  were  brittle, 
giving  low  values  of  Tm  from  about  700  pounds  per  square 
inch  to  less  than  6000  pounds  per  square  inch ;  those  alloys 
failed  at  the  elastic  limit  with  a  total  angle  of  twist  of  only 
i  to  2  degrees. 

Similar  results  with  like  erratic  variations  were  found  by 
Professor  Thurston  for  alloys  of  copper  and  zinc.  The 
greatest  values  of  Tm  ran  from  about  35,000  to  52,000 
pounds  per  sq.  in.  for  90,58%  Cu  with  9.42%  Zn  to  49.66% 
Cu  with  50.14%  Zn. 

It  should  be  observed  that  the  test  specimens  used  by 
Prof.  Thurston  were  .625  inch  in  diameter  with  a  torsion 
length  of  i  inch  only  and  they  were  tested  in  his  torsion 
machine. 

TABLE    I. 
ALUMINUM   ALLOYS— TORSIONAL   RESISTANCE. 


Composition  Per  Cent. 

Angle  of 
Torsion  Deg. 

Torsive  Shear 
per  Sq.In. 

Al. 

Sn.           Cu. 

Elastic 

Maxi- 

Elastic 

Maxi- 

General Character. 

Limit. 

mum. 

Limit. 

mum. 

•  



IOO 

2 

130 

4.300 

25,000 

2.5 

2.5 

95  ' 

4 

200 

10,710 

33,075 

Very  soft;  ductile. 

2.  75 

3-75 

92.5 

6 

198 

11,827 

35,802 

Soft;  ductile. 

5 

5 

90 

7 

175 

15,525 

45,155 

Slightly  tough;   ductile. 

6.25 

6.25 

87-5 

4 

37 

30,282 

63,440 

Tough;   medium  ductility. 

7.5 

7-5 

85 

3-5 

22 

25,447 

37,o62 

Very  tough;   rather  hard. 

8.75 

8.75 

82.5 

7 

10 

18,413 

18,413 

Hard;   somewhat  brittle. 

10 

10 

80 

6 

8 

15,230 

15,230 

Very  hard;  brittle. 

*n 

n 

78 

5.8 

5.8 

13,717 

13,717 

Very  hard;  exceed  'gly  brittle 

*20 

20 

60 

i 

i 

2,321 

2,321 

Very  hard;  ejiceed'gly  brittle 

Scattering. 


2 

IO 

88 

3 

147.5 

14,000 

43,987 

Somewhat  soft;  ductile. 

10 

I 

89 

5 

52 

21,740 

50,000 

Tough;  medium  ductility. 

12 

2 

86 

9 

9 

32,984 

32,984 

Very  tough;  hard. 

13 

2 

85 

8 

12 

32,723 

37,003 

Very  tough;   hard. 

85 

7-5 

7-5 

3 

37 

8,703 

17,630 

Very  soft;   somewhat  ductile. 

27-1 

119 

69.6 

2-5 

2O 

2,800 

2,800 

Very  soft;  spongy. 

IOO 

2 

160 

4,005 

12,911 

*Could  not  be  machined. 


Art.  38.)         APPLICATIONS   OF  FORMULA  FOR   TORSION.  195 


Table  I  contains  experimental  values  of  the  elastic 
limit  and  ultimate  torsion  shearing  resistance  of  the  alloys 
of  aluminum,  tin,  and  copper  shown  in  the  table.  They 
were  determined  by  Messrs.  Gebhardt  and  Ward  in  the 
mechanical  laboratory  of  Sibley  College  at  Cornell  Uni- 
versity and  reported  to  the  Am.  Soc.  Mech.  Engrs.  in 
1898. 

The  results  of  the  table  show  that  the  alloys  yielding 
other  'resistances  of  considerable  value  will  also  exhibit 
proportionate  torsion  resistances,  as  might  be  anticipated. 

The  Eighth  Report  to  the  Alloys  Research  Committee 
of  the  Institution  of  Mechanical  Engineers  of  Great  Britain 
by  Prof.  H.  C.  H.  Carpenter,  M.A.,  Ph.D.,  and  Mr.  C.  A. 
Edwards  in  1907  contains  some  interesting  torsion  tests  on 
specimens  of  copper-aluminum,  the  pieces  being  .624  inch 
in  diameter  and  3  inches  in  length  with  the  exception  of 
No.  3,  which  was  2.8  inches  in  length.  Table  II  gives  the 
results  of  these  tests.  It  will  be  observed  that  alloys  with 
a  comparatively  small  percentage  of  aluminum  give  much 
higher  torsional  ductility  than  pure  copper.  This  is  proba- 
bly due  to  the  fact  that  rolled  copper  generally  contains 

TABLE   II. 


Greatest  -Twisting  Moment 

Cu. 
Per  cent. 

Al. 
Per  cent. 

and  Stress. 

Twist  on 
Whole  Length 
Degrees. 

Ratio, 
Torsion  (Tm) 

Moment. 

Stress,  Lbs. 

Tension 

In.-Lbs. 

per  Sq.In. 

99.96 

0 

1,792 

37,500 

2,736 

•51 

99  9 

.1 

2,293 

47,960 

5,184 

•15 

98.94 

I.  O6 

2,359 

49,350 

4,345 

4i 

97-9 

2.  I 

2,464 

51,450 

3,600 

.  34  

95-95 

4-05 

2,813 

58,870 

2,316 

.2 

93-23 

6-73 

3,306 

69,170 

1,623 

.18 

92.61 

7-35 

3,373 

70,580 

i,374 

•15 

90.06 

9-9 

3,351 

70,IIO 

234 

0.89 

88.2 

ii  .72 

3,584 

74,970 

5i 

1.04 

196  TORSION.  [Ch.  Ill 

some  dissolved  oxygen  which  diminishes  its  ductility.  The 
addition  of  a  small  amount  of  aluminum  removes  the  oxy- 
gen and  enhances  the  ductility.  -The  authors  of  the  report 
express  the  conclusion  that  "Alloys  containing  aluminum 
up  to  7 1  per  cent  behave  extremely  well  under  the  torsion 
test  but  beyond  this  percentage  there  is  a  rapid  deteriora- 
tion of  properties."  The  ratio  between  the  ultimate  resist- 
ance Tm  to  torsional  shear  and  the  ultimate  tensile  resist- 
ance is  shown  in  the  last  column  of  the  table. 

• 
Other  Sections  than  Circular. 

The  common  theory  of  torsion  is  correct  only  for  cir- 
cular sections.  The  general  demonstration  for  other  sec- 
tions than  circular  shows  that  for  square,  rectangular, 
triangular  and  elliptical  sections,  the  maximum  intensity 
of^torsive  stress  Tm  will  be  found  at  the  middle  point  of  a 
side  of  a  square  section  or  of  the  longest  side  of  a  rectangular 
section,  or  at  the  middle  point  of  the  side  of  an  equilateral 
triangular  section  and  at  the  extremities  of  the  minor  axis 
of  an  elliptical  section.  If,  however,  for  approximate  pur- 
poses the  formulae  of  the  common  theory  of  torsion  should 
be  used  for  the  sections  indicated  above  the  polar .  moments 
of  inertia  I9  would  be  taken  from  eqs.  (6),  (7),  (8)  and  (9) 
of  Art.  37.  The  maximum  torsive  shear  Tm,  in  this  pro- 
cedure, should  be  taken  as  existing  at  the  extreme  points 
of  the  section.  The  results  by  this  approximate  method 
will  be  sufficiently  near  for  most  ordinary  purposes,  at  least 
with  the  square  section,  but  the  exact  theory  should  be 
used  for  oblong  sections  or  where  the  highest  degree  of 
accuracy  is  desired  for  non-circular  sections. 


CHAPTER    IV. 
HOLLOW  CYLINDERS  AND  SPHERES 

Art.  39. — Thin  Hollow  Cylinders  and  Spheres  in  Tension. 

If  a  straight  closed  hollow  cylinder  be  subjected  to  an 
interior  pressure  having  the  intensity  q'  sufficiently  greater 
than  that  of  the  exterior  pressure  qit  there  will  be  a  ten- 
dency to  split  the  cylinder  longitudinally. 

Fig.  i  represents  such  a  cylinder  with  sides  so  thin  that 
the  stress  to  which  they  are  subjected  may  be  considered 
uniformly  distributed  throughout 
any  diametral  section.  If  a  cy- 
lindrical shell  has  much  thickness 
relatively  to  its  interior  radius 
the  tensile  annular  stress  due  to 
inner  pressure  will  not  be  uni- 
formly distributed  throughout 
the  shell.  The  excess  of  inner 
pressure  over  the  outer,  if  the  FIG.  i. 

latter  exists,  will  cause  the  inside 

part  of  the  annular  section  of  metal  to  be  stressed  to  a 
higher  intensity  than  the  outside  and  that  difference  will  be 
greater  as  the  thickness  of  the  shell  increases  relatively  to 
the  radius.  It  becomes  necessary  therefore  to  distinguish 
between  these  two  classes  of  cylindrical  shells  in  their  ana- 
lytic treatment. 

AB  represents  the  diametral  plane  through  the  axis  of 
the  cylinder,  the  thickness  i  of  the  shell  being  supposed  in 
this  case  to  be  so  small  that  the  cylindrical  shell  may  be 
considered  "  thin." 

197 


198  HOLLOW   CYLINDERS  AND  SPHERES.  [Ch.  IV. 

As  the  notation  shows  r'  is  the  interior  radius  and  r\ 
the  exterior  radius.  If  C,  the  centre  of  the  cylindrical 
section,  be  taken  as  the  origin  of  the  circular  coordinates 
r'  and  a,  and  if  a  unit  length  of  cylinder  be  considered, 
the  indefinitely  small  amount  of  pressure  on  a  differential 
of  the  interior  surface  r'a  will  be  q'r'da  and  it  will  have  a 
component  at  right  angles  to  the  diametral  plane  AB 
expressed  by  q'r'da  sin  a.  The  integral  of  this  expres- 
sion between  180°  and  o  will  be  the  total  normal  pressure 
acting  on  the  two  longitudinal  sections  of  metal  at  A  and 
B,  as  shown  by  the  following  equation: 


(   q'r'  sin  ada  =  2q'r'. 


One-half  of  the  second  member  of  this  equation,  qY, 
represents  the  tendency  to  split  the  cylinder  at  either  A 
or  B  and  it  must  be  resisted  by  the  sections  of  metal  at 
those  two  points,  or  at  any  other  two  points  at  the  extrem- 
ities of  a  diameter. 

Precisely  the  same  integration  made  for  the  exterior 
pressure  will  obviously  give  the  quantity  q\r\  representing 
the  tendency  to  give  the  metal  compression  at  the  extremi- 
ties of  any  diameter. 

The  resultant  tendency  to  split  the  cylinder  per  unit  of 
length  will  then  be  qfrf  —  q\r\,  it  being  supposed  that  the 
interior  pressure  is  so  much  greater  than  the  exterior  that 
tension  only  will  be  induced  in  the  material.  Obviously 
if  the  exterior  pressure  were  much  larger  than  the  interior, 
compression  would  exist  instead  of  tension.  The  intensity 
of  tensile  stress  t  in  the  sides  of  the  cylinder  will  therefore  be 


Art.  39.]  THIN  HOLLOW  CYLINDERS  AND  SPHERES  IN  TENSION.    199 

This  value  of  t  expresses  the  tendency  of  the  cylinder 
to  split  along  a  diametral  plane  under  the  action  of  the 
interior  pressure  q'  '. 

If  the  ends  of  the  cylinder  are  closed,  the  internal 
pressure  against  them  will  tend  to  force  them  off  or  to  pull 
the  cylinder  apart  around  a  section  normal  to  the  axis. 
The  force  F  tending  to  produce  this  result  will  be 

F-KteV'-^O  ......     (2) 

The  area  of  normal  section  of  the  cylinder  will  be 
n(r^  —  r'2).  Hence  the  intensity  of  stress  developed  by 
this  force  will  be 


r,2-/2 


If  the  exterior  pressure  is  so  small  that  it  may  be  con- 
sidered zero,  eqs.  (i)  and  (3)  give 

t  =  ^f,  (4) 


When  the   thickness  of  the   shell  is  small  /  may  be 
ced  ec 
will  give 


r '  +  r 
placed  equal  to -,  and  this  value  introduced  in  eq.  (5) 


f  in  eq.  (6)  is  seen  to  be  but  half  as  much  as  t  in  eq.  (4). 
In  this  case,  therefore,  if  the  material  has  the  same  ulti- 
mate resistance  in  both  directions,  the  cylinder  will  fail 
longitudinally  when-  the  interior  intensity  is  only  half 
great  enough  to  produce  transverse  rupture. 

In  designing  thin  cylinders  it. will  usually  be  necessary 
to  determine  the  thickness  i,  so  that  the  tensile  stress  t  in 


200  HOLLOW  CYLINDERS  AND  SPHERES,  [Ch.  IV. 

the  metal  shall  not  exceed  the  prescribed  value  h.  After 
writing  h  for  t  in  eq.  (i),  also  r^  —  r'  for  i,  then  dividing 
both  sides  of  the  equation  by  r't  there  will  result 


This  equation  readily  gives 


If  the  exterior  pressure  ql  is  so  small  that  it  may  be 
considered  zero,  the  thickness  given  by  eq.  (7)  takes  the 
following  form  : 


This   is   the    same  value   that  will   be   found   by  solving 
eq.  (4)  for  *. 

The  expression  for  the  thickness  of  the  material  of  the 
cylinder  to  -resist  the  longitudinal  tension  having  the  in- 
tensity /  can  be  found  with  equal  ease.  If  fi  be  written 
for  /  in  eq.  (3),  as  the  greatest  permissible  longitudinal 
tension,  then  if  both  numerator  and  denominator  of  the 
second  member  of  that  equation  be  divided  by  r'2,  there 
will  result 


*"     A 


The  solution  of  this  equation  at  once  gives  the  desired 
thickness : 

/i    _1_  nf\  4 

-r' (9) 


Art.  39.]  THIN  HOLLOW  CYLINDERS  AND  SPHERES  IN  TENSION.    201 

If  q±  is  so  small  that  it  may  be  neglected,  it  is  simply  to 
be  made  zero  in  eq.  (9). 

If  the  exterior  pressure  q^  were  considerably  larger  than 
q',  the  resulting  stresses  in  the  sides  of  the  cylinder  would 
be  compression,  but  the  formulae  for  the  resulting  intensi- 
ties would  be  precisely  the  same  as  the  preceding,  as  long 
as  the  cylinder  retained  its  circular  shape. 

The'  case  of  stresses  in  a  thin  hollow  sphere  or  thin 
spherical  shell  may  be  treated  in  the  same  general  manner. 
The  hemispherical  ends  of  a  metallic  cylindrical  tank  or 
reservoir  may  be  illustrated  by  the  skeleton  section  in 
Fig.  2. 


c 

ji 

FIG.  2. 

As  indicated  in  the  figure  the  internal  radius  of  each 
end  is  r',  while  r\  is  the  external  radius.  The  internal  and 
external  intensities  of  pressure  are  as  shown  in  Fig.  i. 
The  force  tending  to  tear  off  the  hemispherical  ends  of  the 
tank  along  the  line  AB,  Fig.  2,  is  Tr(qr'2—qri2).  The  sec- 
tion of  metal  resisting  this  force  with  the  intensity  /  is 
7r(ri2—  r'2).  The  intensity  of  stress  developed  in  the  metal 
will  therefore  be 

(10) 


r\ 


2  _r>2 


If  the  external  pressure  is  so  small  that  there  may  be 
taken  qi  =o,  eq.  (10)  will  take  the  form 


r\ 


2  _ 


qY 

21 


202  HOLLOW  CYLINDERS  AND  SPHERES.  [Ch.  IV. 

In  this  last  equation  i=n  —  r' ,  and  the  interior  radius 
is  placed  equal  to  one-half  the  sum  of  the  interior  and  ex- 
terior radii,  as  may  be  done  without  sensible  error.  The 
interior  radius  being  given,  the  thickness  of  metal  required 
to  withstand  a  given  internal  pressure  qr  without  stressing 
the  metal  above  a  given  working  value  t  may  be  written 
as  follows  from  eq.  (n) : 


If  the  value  of  the  thickness  i  should  be  desired  in 
terms  of  both  the  interior  and  exterior  pressures,  it  can 
easily  be  written  by  the  aid  of  eq.  (10)  ;  if  both  numerator 
and  denominator  of  the  second  member  of  that  equation 
be  divided  by  r'2,  there.  may  at  once  be  found 

ri_t+g'\* 


After  multiplying  this  equation  through  by  rf,  then  sub- 
tracting that  quantity  from  each  side  of  the  resulting 
equation,  the  desired  value  of  the  thickness  will  be 


By  giving  a  proper  working  value  to  the  tensile  in- 
tensity t  and  inserting  the  values  of  the  pressures,  the  thick- 
ness i  will  at  once  result. 

In  all  these  equations  no  allowance  is  made  for  the 
metal  taken  out  by  the  rivet  holes  in  riveted  work.  This 
does  not,  however,  affect  in  any  way  the  equations  found. 
It  is  only  necessary  to  remember  that  the  cross-section  of 
metal  required  by  the  preceding  equations  is  to  be  regarded 
as  the  net  section,  i.e.,  the  section  remaining  after  the  rivet 


Art.  40.] 


THICK  HOLLOW  CYLINDERS. 


203 


holes  have  been  made.  This  is  equivalent  to  making  the 
thickness  i  great  enough  to  give  the  required  section  as 
net  section. 

Art.  40.    Thick  Hollow  Cylinders. 

If  the  thickness  of  sides  or  walls  of  hollow  cylinders 
and  spheres  subjected  to  high  internal  pressures  is  great 
in  comparison  with  the  internal  radius,  the  tensile  stress 
in  the  metal  may  not  be  assumed  to  be  uniformly  distrib- 
uted, and  it  is  necessary  to  deter- 
mine entirely  different  formulae 
from  those  established  in  the  pre- 
ceding article. 

The  normal  section  of  a  thick 
hollow  cylinder  is  shown  in  Fig. 
i ,  rr  being  the  internal  radius  and 
r\  the  external,  with  the  intens- 
ities of  internal  and  external 
pressures  p'  and  p\  respectively. 
It  is  supposed  that  the  internal 
pressure  so  greatly  exceeds  the 
external  that  the  metal  sustains 
tensile  stress  only.  If  the 
cylinder  be  supposed  to  be 

divided  into  a  great  number  of  thin  concentric  portions, 
the  elastic  stretching  of  the  metal  will  cause  a  much  higher 
tension  to  exist  in  the  interior  portions  than  in  the  exterior. 
If  any  diametral  section,  such  as  AB,  Fig.  i,  be  assumed, 
it  is  clear  that  the  sum  of  all  the  tensile  stresses  developed 
in  that  section  must  be  equal  to  the  excess  of  the  internal 
pressure  over  the  external.  A  unit  length  of  cylinder  will 
be  considered  in  the  following  formula. 

The  tensile  stress  in  the  sides  of  the  cylinder,  whose 
intensity  will  be  represented  by  h,  and  which  is  developed 


204  HOLLOW  CYLINDERS   AND  SPHERES.  [Ch.  IV. 

in  any  diametral  section,  as  AB,  has  a  circumferential 
direction,  and  for  that  reason  it  is  sometimes  called  "  hoop 
tension." 

The  variation  of  this  tensile  intensity  h  carries  with  it 
a  corresponding  variation  in  intensity  of  the  radial  pres- 
sure whose  intensity  is  p,  having  the  values  pf  in  the  in- 
terior of  the  cylinder  and  pi  at  the  external  surface. 

The  amount  of  tension  on  a  radial  section  of  thickness 
dr  will  be  hdr,  and  if  that  differential  expression  be  inte- 
grated so  as  to  extend  over  the  entire  thickness  of  one  wall 
or  side  of  the  cylinder,  it  must  be  equal  to  the  effort  of  the 
internal  pressure  in  excess  of  the  external  to  split  the 
cylinder  along  one  of  its  sides.  The  following  equation 
is  the  analytical  expression  of  this  condition  : 


p'r'  -piri=        dr  ......     (i) 


If  p',  pi,  rf,  and  r\  be  considered  variable  so  as  to  refer 
to  any  interior  points  in  the  wall  of  the  cylinder,  and  if  r' 
and  r\  become  so  nearly  equal  to  each  other  that  r\—r' 
may  be  considered  as  dr,  then  will  p'r'  —  p\r\  =d(pr)  and 
eq.  (i)  will  become: 

d(pr)=pdr+rdp=hdr.        '.     .     .         (2) 

Eqs.  (i)  and  (2)  will  be  in  no  way  changed  if  the  ends 
of  the  cylinder  are  closed,  it  being  assumed  in  that  case 
that  the  longitudinal  stress  is  uniformly  distributed  over  a 
normal  section  like  that  shown  in  Fig.  i. 

Eq.  (2)  is  a  differential  equation  expressing  a  relation 
between  the  two  intensities  p  and  h.  Another  equation  of 
condition  is  required  in  order  to  determine  the  two  unknown 
quantities.  This  second  equation  can  be  written  by  ex- 
pressing the  relation  existing  between  the  direct  and  lateral 


Art.  40.]  THICK  HOLLOW  CYLINDERS.  205 

strains  due  to  the  stresses  p  and  h,  so  as  to  leave  the  radial 
longitudinal  sections  of  the  walls  of  the  cylinder  plane  under 
the  conditions  of  stress  due  to  the  assumed  internal  and 
external  pressures.  The  establishment  of  such  an  equation, 
however,  will  lead  to,  or  express,  precisely  the  same  con- 
ditions involved  in  the  analysis  of  Art.  5  of  Appendix  I, 
which  therefore  need  not  be  repeated  here.  Those  con- 
ditions may  be  expressed  by  stating  that  the  sum  of  the  two 
intensities  p  and  h,  i.e.,  (p+h),  is  a  constant  for  given 
intensities  of  pressure.  If  therefore,  a  be  such  a  constant 

there  will  be  assumed  the  equation: 

\ 
P+h=a (3) 

dp  =  —  dh,  and  p  =  a  —  h. 

By  the  aid  of  these  expressions  eq.   (2)  will  take  the 
form: 

2hdr-\-rdh=adr. 

By  multiplying  both  sides  of  this  equation  by  r  there 
will  result : 

Jf    97  \          aj    9  /     \ 

d(r2h)  =-dr2 (4) 

2 

If  6  is  a  constant  the  integration  of  eq.  (4)  will  give 

/      a  ,  b 

*    I"1^     •••••••     (5) 

Also 

p=a-h  =  2-b.. (6) 

The  interior  and  exterior  pressures  p1  and  pi  are  known, 
and  eq.  (6)  will  give  the  two  equations : 

-      L  a      b 

>i=--—* (7) 


206  HOLLOW  CYLINDERS  AND  SPHERES.  [Ch.  IV. 

By  subtracting  pi  from  pf 


(8) 


Then  by  the  second  of  eqs.  (7)  : 

a  b      p'r'2-pin2 

- 


...      (9) 


The  substitution  of  these  values  of  b  and  -  in  eqs.  (5) 

and  (6)  will  give  the  following  values  of  the  intensities  p 
and  h.  Inasmuch  as  the  preceding  equations  involving  h 
and  p  have  been  written  without  giving  distinctive  signs 
to  either  tension  or  compression  and  as  the  constants  b 

and  —  may  be  regarded  either  as  positive  or  negative,  the 
2 

sign  of  each  one  will  be  changed  by  writing  ri2—  r'2  for 
r'2  —  n2,  which  will  make  the  tensile  stress  h  positive  and 
the  compressive  stress  p  negative  after  substituting  the 

values  of  the  constants  b  and  -  in  eqs.  (5)  and  (6). 

2 


h=<"  2  ^  x  +^2  V  I1.  .  .  .  (n) 

n2_f'2  fl2_f'2       f2 

Eqs.  (10)  and  (u)  can  be  put  in  more  convenient  form 
for  use  in  numerical  computations  by  dividing  both  numer- 
ator and  denominator  of  all  the  terms  in  the  second  mem- 
bers of  those  equations  by  n2.  This  simple  operation  will 
give  eqs.  (12)  and  (13): 


Art.  40.]  THICK  HOLLOW  CYLINDERS.  207 


'2 


Eqs.  (12)  and  (13)  are  the  general  values  of  the  inten- 
sities of  the  internal  stresses  in  the  walls  of  the  cylinder, 
p  acting  in  a  radial  direction  and  h  in  a  circumferential 
direction.  The  greatest  tensile  intensity  h'  will  exist  at 
the  interior  surface  of  the  cylinder  and  it  will  be  found 
by  making  r  =  r'  in  eq.  (13)  as  shown  by  eq.  (14)  : 

'2 


Similarly  the  intensity  of  tensile  stress  at  the  outer 
surface  of  the  cylinder  (the  least  intensity  of  tensile  stress) 
will  be  given  by  making  r  =  r\. 

'2  r'2\ 

y 

.....    ds) 


The  thickness  t  of  the  wall  of  the  cylinder  which  must 
be  provided  if  the  greatest  intensity  of  tensile  stress  hf  is  not 
to  be  exceeded  by  a  given  intensity  p'  of  interior  pressure, 


208  HOLLOW  CYLINDERS  AND  SPHERES.  [Ch.  IV. 

can  readily  be  found  by  solving  eq.  (14)  for  the  quantity 

r'2   ' 

—5,  which  will  give  eq.  (16). 

rr 

h' 


Then  by  adding  (-1)  to  each  side  of  eq.  (16)  and 
multiplying  both  members  by  r'  eq.  (17)  will  at  once 
result  : 

'.  -  -  •  <•» 


As  the  internal  radius  r'  will  always  be  known,  eq.  (17) 
gives  the  thickness  t  desired  in  terms  of  the  known  pres- 
sures and  the  intensity  of  working  stress  hf. 

Eq.  (17)  shows  that  if  2pi+h'=pf,  t  will  be  infinity. 
This  shows  that  when  the  intensity  of  the  internal  pressure 
is  equal  to  or  greater  than  twice  the  intensity  of  the  exter- 
nal pressure  added  to  the  greatest  allowed  tensile  stress  in 
the  metal,  it  is  impossible  to  make  the  wall  of  the  cylin- 
der thick  enough  to  resist  that  internal  pressure. 

If  the  external  pressure  is  so  small  that  it  may  be 
neglected,  it  is  necessary  only  to  place  pi=o  in  the  pre- 
ceding equations. 

If  pi  exceeds  p'  it  is  obvious  that  the  internal  stress  h 
will  be  compression,  i.e.,  there  will  be  hoop  compression 
as  the  circumferential  stress  in  the  cylinder  wall  instead 
of  hoop  tension. 

The  complete  solution  of  the  problem  of  the  thick 
cylinder  including  expressions  for  the  distortions  or  strains 
of  the  material  at  all  points  will  be  found  in  Art.  5  of 
Appendix  I. 

The  application  of  the  preceding  formulae  can  be  ex- 
pedited by  the  use  of  the  following  tabular  values  which 


Art.  40.] 


THICK  HOLLOW  CYLINDERS. 


209 


explain  themselves.  A  curve  more  useful  than  the  table 
can  readily  be  constructed  from  the  numerical  values  in 
the  latter,  so  that  any  value  whatever  for  the  ratio  of  the 
radii  indicated  can  be  read  at  sight. 


r' 
r 

r'2 

n* 

r' 
r 

r'2 
r»« 

I. 

I. 

•  5 

•25 

•95 

.9025 

•45 

.2025 

•9 

.81 

•4 

.16 

•85 

.7225 

•35 

.1125 

.8 

.64 

•3 

.09 

•75 

•5625 

•25 

.0625 

•7 

•49 

.2 

.04 

•65 

•4225 

•15 

.0225 

.6 

•36 

.1 

.01 

•55 

•3025 

•05 

.0025 

As  an  illustration  of  the  laws  of  variation  of  the  inten- 
sities h  and  p  the  following  data  may  be  used: 

r'  =  10  inches; 

h'  =  20,000  pounds  per  square  inch; 

p'  =  10,000  pounds  per  square  inch; 

pi  =  1000  pounds  per  square  inch. 

Eq.  (17)  will  give,  after  a  substitution  in  it  of  the  above 
numerical  data,  £  =  5.81  inches. 

ri  =  15.81  inches. 

r>2 

The  quantity  —  will  have  values  ranging  from  unity 

for  the  interior  of  the  cylinder  to  --  =.4.     Inserting  these 

values  in   eqs.    (12)    and    (13)    there   will   result   the   two 
equations : 


=  5000-15,000  — ; 


210 


HOLLOW  CYLINDERS  AND  SPHERES. 


[Ch.  IV. 


h  =  5000  +  15,  ooo  —  . 


Taking  the  varying  values  of  —   given  in  the  above 
table  the  following  values  of  p  and  h  will  result  : 


Pounds  per  Sq.in. 

r' 

r 

P 

b 

!_ 

—  IO,OOO 

20,000 

•95 

-  8,540 

18,540 

•9 

-  7,150 

17,150 

•85 

-  5,840 

15,840 

.8 

—  4,600 

14,600 

•75 

-  3,440 

13,440 

•  7 

-  2,350 

12,350 

•65 

-  1,340 

ii,340 

•63 

—  1,000 

11,000 

Fig.  2  represents  these  results  graphically.     The  straight 
line  GAP  is  laid  off  tangent  at  any  point  A  to  the  circle 

HB  D 


FIG.  2. 


representing  the  interior  of  the  cylinder  subjected  to  the 
pressure  of  10,000  pounds  per  square  inch.     Similarly  HBD 


Art.  40.]  THICK  HOLLOW  CYLINDERS.  211 

is  a  straight  line  laid  off  tangent  at  the  point  B  on  any  radius 
CB  of  the  exterior  surface  of  the  cylinder,  the  distance  A  B 
being  equal  to  5.81  inches.  AF  is  then  laid  off  by  scale 
equal  to  h'  =  20,000  pounds,  while  A G  is  similarly  laid  off  to 
represent  p'  =  10,000  pounds  per  square  inch,  but  it  must  be 
remembered  that  it  acts  in  a  radial  direction,  i.e.,  along 
A  B.  BD  and  BH  are  the  corresponding  quantities  for  the 
exterior  surface  of  the  cylinder,  equal  respectively  to  11,000 
pounds  and  1000  pounds.  Curves  DF  and  HG  are  then 
constructed  by  laying  off  the  ordinates  p  and  h  at  right 
angles  to  A  B  as  shown. 

Case  of  Exterior  Pressure  Greater  than  Interior  Pressure. 

If  the  exterior  pressure  pi  is  greater  than  the  interior 
pressure  p' ',  it  is  evident  that  the  preceding  equations  will 
need  no  change  whatever,  but  the  difference  p'  —  pi  will 

r'2 
now  be  negative.     As  p' — -  is  less  than  p' ,  p  will  still  be 

negative  and  represent  compression.  On  the  other  hand, 
h  will  now  be  negative  and  represent  circumferential  or 
hoop  compression  as  shown  by  eq.  (13).  Eqs.  (12)  and 
(13)  are  used  in  connection  with  this  case  in  designing 
modern  heavy  guns  where  thick  cylinders  are  raised  to  a 
high  temperature  and  slipped  over  a  close-fitting  interior 
thick  cylinder  at  ordinary  temperature,  so  that  when  the 
outside  hot  cylinder  cools  it  contracts  and  puts  the  interior 
cylinder  under  a  high  compression.  In  fact,  the  lining  of 
the  gun  may  be  enclosed  by  two  or  more  such  cylinders 
successively  shrunk  into  place.  One  interior  cylinder  with 
slightly  conical  interior  surface  may  be  forced  by  a  high 
pressure  at  ordinary  temperature  into  the  interior  of  a 
corresponding  exterior  cylinder  with  similar  results.  These 
matters  will  be  treated  more  extendedly  in  the  next  article. 


212  HOLLOW   CYLINDERS  AND  SPHERES.  [Ch.  IV. 

Art.  41.  —  Radial  Strain  or  Displacement  in  Thick  Hollow  Cylin- 
ders. —  Stresses  Due  to  Shrinkage  of  One  Hollow  Cylinder 
on  Another. 

Radial  Strain  or  Displacement. 

Inasmuch  as  all  diametral  sections  of  thick  hollow 
cylinders  remain  plane  for  all  conditions  of  stress  due  to 
internal  or  external  pressure,  the  only  strain  or  displace- 
ment in  such  a  cylinder  is  that  in  a  radial  direction  due  to 
either  increase  or  decrease  of  the  diameter  of  any  elementary 
thin  cylinder  or  shell  with  radius  r.  This  radial  displace- 
ment will  be  indicated  by  p  and  the  expression  for  it  can 
only  be  established  by  the  analysis  shown  in  Art.  5  of 
Appendix  I,  or  by  some  equivalent  analysis.  By  referring 
to  eq.  (10)  and  the  two  equations  preceding  eq.  (15)  of  that 
article  it  will  be  seen  that  the  desired  displacement  is  given 
by  the  following  equation: 


"2 


In  this  equation  G  is  the  modulus  of  elasticity  for  shear- 
ing, while  pi  and  r\  represent  the  intensity  of  exterior 
pressure  and  the  exterior  radius,  respectively,  and  p'  and 
r'  similar  quantities  for  the  interior  of  the  cylinder.  The 
quantity  r  represents  the  ratio  of  the  lateral  strain  divided 
by  the  direct  strain,  i.e.,  Poisson's  ratio.  Obviously  if 
r=r' ',  the  increase  or  decrease  of  the  interior  radius  will  be 
given  by  p  and  a  similar  observation  applies  to  the  increase 
or  decrease  of  the  exterior  radius  when  r  =  r\. 

It  is  clear  that  if  r  be  made  equal  to  either  r'  or  r\  in 
eq.  (i)  either  p'  or  pi  may  be  written  from  that  equation 


Art.  41.]  STRESSES  DUE   TO  SHRINKAGE.  213 

in  terms  of  the  corresponding  radial  displacement  p'  or  pi. 
It  is  sometimes  desirable  to  express  the  intensities  of 
interior  or  exterior  pressures  in  this  manner,  after  having 
determined  the  radial  displacement  corresponding  to  a 
known  change  of  temperature  or  in  some  other  manner. 
In  the  operation  of  shrinking  one  cylinder  on  another  the 
difference  in  diameters  required  for  the  operation  may  be 
prescribed  by  some  empirical  rule. 

Stresses  Due  to  Shrinkage. 

It  has  been  shown  in  the  preceding  article  that  when  a 
thick  hollow  cylinder  is  subjected  to  a  high  internal  pressure 
the  intensity  of  circumferential  or  hoop  tension  is  much 
greater  at  and  near  the  interior  surface  of  the  cylinder  than 
at  the  exterior  surface  and  that  if  the  thickness  is  great 
the  intensity  of  the  interior  tension  may  be  high,  while 
that  of  the  exterior  surface  will  be  extremely  low,  show- 
ing the  use  of  the  metal  to  be  uneconomical.  In  heavy 
gun  making  this  undesirable  condition  is  overcome  by 
dividing  the  body  of  the  gun  into  a  number  of  concentric 
thick  cylinders,  each  being  shrunk  over  those  inside  of  it, 
after  making  the  interior  diameter  at  ordinary  temperature 
less  than  the  exterior  diameter  of  that  over  which  it  is 
shrunk  into  place.  Each  tube  is  heated  so  as  to  enlarge 
its  diameter  until  it  can  be  slipped  over  the  tube,  or  tubes, 
inside  of  it,  so  that  when  it  cools  it  will  itself  be  subjected 
to  high  internal  pressure  with  correspondingly  high  cir- 
cumferential or  hoop  tension,  while  the  tube,  or  tubes, 
inside  of  it  will  be  correspondingly  compressed  at  ordinary 
temperature.  The  body  of  the  gun  thus  composed  of  a 
series  of  concentric  tubes  shrunk  in  place  in  series  will  form 
a  combination  in  the  interior  of  which  there  will  be  rela- 
tively high  circumferential  or  hoop  compression,  decreasing 


214 


HOLLOW  CYLINDERS  AND   SPHERES. 


[Ch.  IV. 


outwardly  though  not  regularly  or  continuously,  with  cir- 
cumferential or  hoop  tension  in  the  outer  part  or  parts. 
When  the  intensely  high  pressures  of  modern  explosives 
are  produced  in  firing  the  gun  the  metal  will  be  more  nearly 
uniformly  stressed  in  circumferential  tension  and  thus  act 
more  effectively  throughout  the  entire  thickness  of  the 
.wall  of  the  gun.  It  will  not  be  attempted  here  to  give 


FIG.  i. 


the  details  required  to  secure  the  best  effects  by  shrinking 
into  place  a  series  of  thick  hollow  cylinders  in  the  manu- 
facture of  ordnance,  but  the  general  analytic  procedure 
in  deducing  the  proper  results  for  the  shrinkage  of  one 
cylinder  on  another  either  in  gun  making  or  in  the  making 
of  other  compound  cylinders  for  high  internal  pressures 
will  be  illustrated  by  a  single  computation  only. 

Fig.    i    represents   a   thick  hollow  steel  cylinder  with 


Art.  4i.j  STRESSES  DUE   TO  SHRINKAGE.  215 

internal  diameter  of  12  inches  and  total  thickness  of  wall 
of  12  inches  composed  of  an  outer  cylinder  6  inches  thick 
shrunk  on  an  interior  hollow  steel  cylinder  with  wall  also 
6  inches  thick.  It  will  be  supposed  that  the  coefficient 
of  expansion  of  steel  per  degree  Fahr.  is  6  =  .0000065.  The 
increase  in  diameter  due  to  a  change  of  225°  Fahr.  of  the 
interior  24-inch  cylinder  will  be  225X24X5  =  . 03 51  inch. 
The  change  in  radius  will  be  one-half  of  this  amount.  The 
interior  diameter  of  the  exterior  thick  cylinder  at  ordinary 
temperature  must  be  24  — .0351  =23.9649  inches. 

If  r"  be  the  interior  radius  of  the  exterior  cylinder 
before  being  heated  and  rH  the  exterior  radius  also  before 
being  heated,  while  r'  and  r\  represent  the  interior  and 
exterior  radii  of  the  interior  cylinder  at  ordinary  tempera- 
ture and  before  shrinkage,  as  shown  in  Fig.  i,  the  data 
required  will  be  as  follows : 

r'=6";    fi=i2A/;    r"  =  11.98245  ;    r/y  =  17.98245. 

The  interior  pressure  of  the  inner  cylinder  will  be  simply 
that  due  to  atmosphere.  Similarly  the  exterior  pressure 
on  the  exterior  cylinder  will  also  be  that  due  to  the  atmos- 
phere. Hence,  both  these  pressures  will  be  considered 
zero.  There  will  then  be  acting  the  shrinkage  pressure  on 
the  exterior  surface  of  the  inner  cylinder  and  the  same 
pressure  on  the  interior  surface  of  the  outer  cylinder.  The 
intensity  of  this  common  shrinkage  pressure  will  be  indi- 
cated by  pi. 

As  indicated  in  Fig.  i,  after  the  properly  heated  outer 
cylinder  has  been  slipped  over  the  inner  cylinder  at  ordi- 
nary temperature  and  the  two  allowed  to  cool,  the  radius 
r\  will  be  decreased  by  the  radial  displacement  pi,  while 
the  radius  r"  will  be  increased  by  the  amount  p".  In- 


2l6 


HOLLOW   CYLINDERS  AND  SPHERES. 


[Ch.  IV. 


asmuch  as  pi  will  be  intrinsically  negative,  eqs.  (2)  will  at 
once  result. 

i  //  i    //  //  //  /  \ 

ri  +  pi=r   +  p       .'.      p    —  p\=r\  —  r  ...     (2) 

By  making  pf  =o  in  eq.  (i)  and  r=r\  there  will  result 
eq.  (3): 


Pi 


(3) 


Similarly  by  making  pi=o  in  eq.  (i),  r'=r",  r\=rlt, 
r=r"  and  remembering  that  p' =  p" =p\,  eq.  (4)  will  repre- 
sent the  increase  of  the  radius  of  the  exterior  cylinder  after 
the  operation  of  shrinkage  is  complete: 


((*-*£ 

//    V          r//2 

P /    //o 


.     .     .     (4) 


By  substituting  the  second  members  of  eqs.  (3)  and  (4) 
for  the  first  member  of  eq.  (2)  an  equation  will  result 
giving  the  value  of  p\  as  shown  by  the  following  equation 
and  eq.  (5) : 


2G 


=  ri-r 


Or 


Hence 


(5) 


Art.  4I-]  STRESSES  DUE  TO  SHRINKAGE.  217 

The  quantity  represented  by  Z  is  clear. 

It  should  be  observed  that  the  relation  between  the 
changes  of  the  radii  at  the  cylindrical  surface  of  shrinkage 
contact  and  the  original  radii  (eq.  (2))  is  general  and  holds 
for  all  conditions  of  shrinkage  stresses  whatever  may  be 
the  thicknesses  of  the  two  cylindrical  walls. 

In  computing  the  value  of  pi  by  eq.  (5)  there  will  be 
taken  : 

£  =  12,000,000  and  r  =  .25. 

If  the  values  already  given  for  the  four  radii  of  the  cylinders 
be  inserted  in  the  equation  immediately  following  eq.  (4), 
there  will  result  : 

Z=-38.4. 
Hence 

•0351  X-5  X24,ooo,ooo  1U 

pi  =—  ^—  -  =  10,970  Ibs.  per  square  inch. 


In  computing  the  value  of  Z  it  is  essentially  accurate 
to  use  the  inner  and  outer  radii  of  the  outer  cylinder  as 
they  exist  before  it  is  heated  for  the  shrinkage  process. 
This  will  save  much  labor  and  simplify  the  application  of 
the  formulae,  but  the  difference  r\—  r"  must  of  course  be 
expressed  as  accurately  as  possible. 

The  stresses  in  the  walls  of  the  two  cylinders  due  to 
shrinkage  may  now  be  readily  computed,  since  the  outer 
cylinder  is  subjected  to  an  inner  pressure  of  10,970  Ibs.  per 
square  inch  and  the  inner  cylinder  to  an  exterior  pressure 
of  the  same  intensity.  The  resulting  values  of  p  and  h 
for  the  two  cylinders  are  as  follows  : 

Inner  Cylinder  in  Compression. 

£1  =  10,970  Ibs.  per  square  inch;  p'  =  o;  rf  =6  inches; 
r\  =  12  inches. 


2l8 


HOLLOW  CYLINDERS  AND  SPHERES. 


ich.  iv. 


Eqs.  (12)  and  (13)  of  the  preceding  article  will  give  at 
once  eqs.  (6)  and  (7)  for  this  case: 


,'2 


I  — 


(6) 


—  I 


r\' 


.'•2 


(7) 


—  I 


If  the  intensities  p  and  h  are  computed  at  six  equidistant 
points  at  the  two  surfaces  and  at  intermediate  points  one- 
fifth  of  the  thickness  of  the  wall  of  the  cylinder  apart,  the 
results  given  in  the  following  tabulation  will  be  found  and 
they  are  shown  graphically  in  Fig.  2. 


Pounds  p 

er  Sq.  In. 

r 

r'« 

r2 

P 

k 

I. 

0 

—  29,250 

.2 

.6944 

-  4-470 

-24.780 

•  4 

.5102 

-  7,164 

—  22,090 

.6 

.3906 

-  8,913 

-20,340 

.8 

.3086 

-10,113 

—  19,089 

2 

•25 

—  10,970 

-18,283 

Outer  Cylinder  in  Tension. 

£'  =  10,970  Ibs.  per  square  inch;  p\=o\  r'  =  i2  inches; 
r\  =  1 8  inches. 

By  making  p\=o  in  eqs.  (12)  and  (13)  of  the  preceding 
article  there  will  result  the  following  two  formulae  for  the 
intensities  p  and  h: 


Art.  41.]       CYLINDER   UNDER  HIGH  INTERNAL  PRESSURE.          219 


,'2 


.'•2 


—  I 


—  I 


(8) 


(9) 


The  two  intensities  p  and  /z-  will  be  computed  for  six 
equidistant  points,  including  those  on  the  two  cylindrical 
surfaces,  by  taking  corresponding  values  of  r.  The  results 
of  these  computations  are  given  in  the  following  tabulation 
and  they  are  shown  graphically  in  Fig.  2,  as  will  be  explained 
further  on. 


r 

r'z 

Pounds  i 

>er  Sq.In. 

r' 

r* 

P 

h 

I. 

—  10,970 

+28,054 

.1 

.2 

.8264 
.6944 

-  7,543 
-  4,937 

+25,093 

+  22,486 

•3 

•5917 

-  2,908 

+20,459 

•4 

.5102 

-  1,299 

+  18,848 

•  5 

•4444 

0 

+  17,552 

Combined  Cylinder  under  High  Internal  Pressure. 

The  stresses  induced  by  shrinkage  in  making  the  com- 
bined cylinder  of  two  concentric  shells  have  been  explained 
and  computed  in  the  preceding  sections;  those  stresses  are 
permanent  and  they  must  be  combined  with  stresses  which 
may  be  produced  usually  temporarily  by  subjecting  the 
combined  cylinder  to  a  high  internal  pressure  such  as  that 
caused  by  the  discharge  of  a  gun.  The  internal  pressure 


220 


HOLLOW  CYLINDERS  AND  SPHERES. 


[Ch.  IV. 


produced  by  a  modern  high  explosive  may  reach  50,000 
or  60,000  Ibs.  per  square  inch,  but  as  an  illustration  in  this 
case  the  internal  pressure  will  be  taken  as  40,000  Ibs.  per 
square  inch.  Hence  the  foUcr.Ting  data  are  required: 

^'=40,000  Ibs.  per  square  inch;    pi=o\    r'  =6  inches; 
r\  =  1 8  inches. 

As  this  case  is  similar  to  that  expressed  by  eqs.  (8)  and 
(9),  those  equations  will  yield  the  results  shown  in  the 
following  tabulation  when  the  above  data  are  substituted 
in  them. 


r 

r'» 

Pounds  per  Sq.In. 

r' 

r* 

P 

h 

I 

I. 

—  40,OOO 

+  50,000 

II 

.5625 
•36 

-20,313 
—  11,196 

+30,312 
+  21,200 

2 

•25 

—  6.2S2 

+  16,248 

2* 

.1837 

-  3,268 

+  13,268 

2f 

.  1406 

-  1,328 

+  11,328 

3 

.1111 

o 

+  10,000 

It  will  be  observed  that  the  intensities  p  and  h  have  been 
computed  at  points  3  inches  apart  throughout  the  12 -inch 
thickness  of  the  combined  cylinder  wall. 

The  results  of  computations  shown  in  the  three  pre- 
ceding tabulations  may  now  be  shown  graphically  in  Fig. 
2.  That  figure  shows  part  of  a  normal  section  of  the  two 
cylinders,  C  being  the  center  and  CD  being  the  internal 
radius  of  6  inches.  The  separate  walls  each  6  inches  thick 
are  shown  by  the  parts  of  concentric  circles  with  radii 
6  inches,  12  inches,  and  18  inches.  The  line  ABD  repre- 
sents the  trace  of  a  longitudinal  diametral  plane  at  right 
angles  to  which  the  intensities  of  the  circumferential  or 
hoop  stresses  shown  in  the  preceding  tabulations  are  laid  off. 


Art.  41.]     CYLINDER   UNDER  HIGH  INTERNAL  PRESSURE.  221 

Tensile  stresses  indicated  by  the  plus  sign  are  laid  off  to 
the  left  of  AD  and  compressive  stresses  to  the  right  of  BD 
as  indicated  by  the  minus  sign. 

Referring  to  the  tabulated  results  for  the  inner  cylinder 
in  compression,  DQ  represents  29,250  and  BP  18,283,  both 
pounds  per  square  inch.  Intermediate  ordinates  of  the 
curved  line  PQ  are  laid  off  by  the  same  scale  to  represent 
the  other  intensities  h  given  in  the  table. 

The  ordinate  AL  represents  by  the  same  scale  the 
intensity  17,552  Ibs.  per  square  inch  and  MB  the  intensity 
28,054  Ibs.  per  square  inch,  both  shown  in  the  tabulation 
for  the  outer  cylinder  in  tension.  The  other  intensities 
laid  off  as  ordinates  give  the  curved  line  LM. 

The  tensile  intensities  h  for  the  combined  cylinder  under 
the  internal  pressure  of  40,000  Ibs.  per  square  inch  are 
shown  by  the  ordinates  to  the  curved  line  EF,  FD  repre- 
senting 50,000  Ibs.  per  square  inch  and  EA  10,000  Ibs.  per 
square  inch. 

The  resultant  intensities  at  various  points  in  the  wall 
of  th*e  combined  cylinder  are  found  by  taking  the  algebraic 
sum  at  each  point  of  the  three  results  shown.  The  result- 
ant hoop  stress  at  D  is  found  by  laying  off  KF=DQ, 
the  resultant  intensity  being  £^  =  50,000  —  29,250  =  20,750 
Ibs.  per  square  inch.  Similarly  BH=MB-BP  =  16,248  - 
18,283  =  — 2035  Ibs.  per  square  inch,  showing  that  the  tensile 
stress  developed  by  the  high  internal  pressure  was  not 
quite  enough  to  overcome  the  shrinkage  compression.  The 
intensities  of  hoop  stress  in  the  wall  of  the  inner  cylinder 
are  therefore  the  intercepts  of  ordinates  at  right  angles 
to  BD  between  FM  and  KH. 

All  stress  in  the  outer  cylinder  is  tension  equal  in 
intensity  at  any  point  to  the  sum  of  the  ordinates  between 
A  B  and  ME  added  to  those  between  A  B  and  LS  repre- 
sented by  the  ordinates  drawn  from  A  B  to  ON.  Thus  it 


222 


HOLLOW  CYLINDERS  AND  SPHERES. 


[Ch.  IV. 


is  seen  that  the  shaded  parts  of  the  diagram  represent  at 
each  point  the  intensity  of  stress  existing  at  that  point. 


o 


The  highest  tension  exists  in  the  outer  cylinder  at  B  and  is 
equal  to.  28,054  +  16,248  =44,302  Ibs.  per  square  inch.     At 


Art.  41.]     CYLINDER   UNDER  HIGH  INTERNAL  PRESSURE.  223 

the  outer  point  A  the  tensile  intensity  of  hoop  stress  is  seen 
to  be  27,552  Ibs.  The  intensity  of  hoop  stress  at  the 
interior  surface  of  the  cylinder  has  been -found  to  be  20,750 
Ibs.  per  square  inch,  materially  less  than  at  the  outer  sur- 
face, which  is  desirable,  as  the  radial  normal  pressure  at 
the  inner  point  is  40,000  Ibs.  per  square  inch. 

The  high  tensile  intensity  44,302  Ibs.  per  square  inch, 
found  at  the  inner  surface  of  the  outer  cylinder  and  the 
compression  of  about  2000  Ibs.  per  square  inch  at  the 
adjacent  point  on  the  inner  cylinder  show  the  desirability 
of  a  redesign  for  the  assumed  internal  pressure  with  adjust- 
ment of  the  amount  of  shrinkage  and  with  the  wall  com- 
posed perhaps  of  three  cylinders  instead  of  two.  In  this 
manner  the  undesired  extremes  of  stress  in  the  vicinity  of 
the  middle  of  the  wall  can  be  avoided.  The  results,  how- 
ever, exhibit  completely  the  procedures  to  be  followed 
where  it  is  desired  to  make  a  combined  cylinder  with  a 
number  of  concentric  shells  with  shrinkage  so  employed  as 
to  produce  a  more  nearly  uniform,  though  not  continuous, 
stress  condition  than  can  be  attained  in  a  single  wall  with- 
out shrinkage.  In  a  single  wall  of  12 -inch  thickness  in 
this  case  the  hoop  tension  would  have  varied  from  50,000 
Ibs.  per  square  inch  at  the  interior  surface  to  only  10,000 
Ibs.  per  square  inch  at  the  exterior  surface. 

The  radial  compressive  intensities  p  have  not  been 
plotted  in  Fig.  2,  as  the  resultant  intensity  in  every  case  is 
found  by  adding  the  intensities  due  to  each  condition  as 
given  in  the  tabulations.  At  the  interior  surface  the  maxi- 
mum intensity  of  pressure  is  40,000  Ibs.  per  square  inch. 
At  3  inches  from  the  interior  surface  the  maximum  inten- 
sity will  be  about  20,000  Ibs.  per  square  inch  and  at  the 
common  surface  of  the  two  shells  that  intensity  will  be 
about  17,000  Ibs.  per  square  inch,  thus  decreasing  outward 
until  the  value  o  is  found  at  the  outer  surface. 


224  HOLLOW  CYLINDERS  AND  SPHERES.  [Ch.  IV. 

Art.  42.— Thick  Hollow  Spheres. 

When  the  thickness  of  wall  of  a  hollow  sphere  is  so 
great  that  the  stresses  may  not  be  considered  uniformly 
distributed  over  a  diametral  section  of  the  shell  the  approxi- 
mate formulae  of  Art.  39  cannot  be  used;  it  becomes  neces- 
sary to  make  an  investigation  similar  to  that  required  for 
thick  hollow  cylinders. 

It  will  be  supposed  that  the  interior  of  the  spherical 
shell  is  subjected  to  an  intensity  of  pressure  p'  greater  than 
the  exterior  normal  pressure  pi  as  shown  in  Fig.  i.  As 
the  intensity  of  the  interior  pressure,  produced  possibly 
by  a  fluid,  is  greater  than  that  of  the  exterior  pressure  the 
material  of  the  shell  will  be  subjected  to  an  internal  stress 
of  tension  as  well  as  the  radial  compression,  but  the  formulae 
as  demonstrated  will  be  equally  applicable  to  the  case  of 
the  exterior  pressure  being  greater  than  the  interior  with- 
out any  modification  whatever.  In  the  latter  case,  however, 
the  internal  stress  acting  around  a  great  circle  will  be  com- 
pression instead  of  tension.  The  formulae  will  be  so  written 
that  a  tensile  stress  is  positive  and  a  compressive  stress 
negative. 

If  a  diametral  section  of  the  spherical  shell  be  taken  as 
in  Fig.  i,  it  is  clear  that  for  a  given  radius  r  there  will  be 
a  uniform  intensity  of  tension  normal  to  that  section  and 
no  other  stress,  i.e.,  this  tension  at  every  point  will  be  in 
the  direction  of  the  circumference  of  a  great  circle.  Fur- 
thermore, since  that  observation  is  true  of  all  possible 
diametral  sections  of  the  shell  it  is  equally  obvious  that  at 
any  point  in  the  shell  there  will  be  two  circumferential 
or  hoop  stresses  at  right  angles  to  each  other  and  a  third 
radial  stress  of  compression  with  no  other  stress  on  its 
surface  of  action,  the  three  stresses  being  principal  stresses 
at  the  assumed  point.  The  three  principal  planes  on  which 


Art.  42-] 


THICK  HOLLOW  SPHERES. 


22$ 


these  principal  stresses  act  are  two  of  them  diametral  and 
at  right  angles  to  each  other,  while  the  third  is  tangent 
to  the  spherical  surface  with  radius  r,  and  it  is  at  right 
angles  to  the  other  two  planes.  The  state  of  stress  in  the 
interior  of  the  shell  is  also  obvious  from  the  fact  that  the 
interior  and  exterior  fluid  or  normal  pressures  are  each  the 
same  in  intensity  at  all  points  making  the  resulting  con- 


FlG.    I. 

dition  of  stress  completely  symmetrical.  As  every  diam- 
etral plane  section  of  the  shell  is  a  principal  plane  of  stress 
there  will  be  no  shear  on  any  such  plane  and  for  the  same 
reason  there  will  be  no  shear  on  any  of  the  concentric 
spherical  surfaces  within  the  limits  of  the  shell. 

Remembering  that  the  interior  radius  of  the  shell  is  r' 
and  the  exterior  radius  r\  and  that  the  tendency  to  tear  the 
shell  apart  in  any  diametral  annular  section  is  due  to  the 
excess  of  the  interior  pressure  over  the  exterior  the  follow- 
ing equation  may  be  at  once  written,  if  h  represents  the 


226  HOLLOW  CYLINDERS  AND  SPHERES.  [Ch.  IV. 

intensity  of  the  internal  tensile  stress  developed  at  any 
point  in  the  annular  section: 


(i) 


If  in  eq.  (i),  r'  and  r\  be  considered  variable  and  of 
so  nearly  the  same  value  that  they  differ  from  each  other 
only  by  dr,  the  quantity  p'r'2  —piri2  becomes  equal  to  d(pr2). 
Hence,  eq.  (i)  for  that  supposition  may  take  the  following 
form: 

d(pr2)  =2hrdr  =  2prdr  +r2dp  .....     (2) 

This  is  a  differential  equation  between  h  and  p. 
Another  equation  of  condition  is  required  to  determine 
those  two  variable  quantities.  Such  an  equation  may  be 
written  by  so  expressing  the  relation  between  the  internal 
distortions  or  strains  accompanying  the  stresses  h  and  p 
as  to  make  the  diametral  sections  of  the  shell  plane  what- 
ever may  be  the  intensities  of  the  internal  stresses  h  and  p. 
The  consideration  of  such  relations  between  the  strains 
produced  would  be  precisely  the  same  as  given  in  Art.  8 
of  Appendix  I,  and  hence  it  is  repeated  here.  If  it  be 
remembered  that  the  intensities  of  the  two  circumferential 
stresses  at  any  interior  point  of  the  shell  are  equal  to  each 
other  and  indicated  by  h,  as  in  eqs.  (i)  and  (2),  while  p 
represents  the  intensity  of  the  internal  radial  stress  at  the 
same  point,  the  relation  between  the  internal  strains  or 
distortions  necessary  to  make  all  diametral  sections  of  the 
shell  plane  for  all  intensities  of  stress  is  equivalent  to  the 
condition  that  the  sum  of  the  three  principal  stresses  must 
be  constant  at  all  points  as  expressed  by  eq.  (3),  a  being 
constant  : 


Art.  42.]  THICK  HOLLOW  SPHERES.  227 

From  eq.  (3): 


p=a  —  2h  and  dp  =  2dh.       ....     (4) 

Substituting   from    eq.    (4)    in    the    second    and    third 
members  of  eq.  (2)  : 

2hrdr  =  2ardr  —  hrdr  —  2r2dh. 


By  arranging  terms  the   preceding  equation  takes  in- 
tegrate form  as  given  by  eq.  (5)  : 

$hrdr+r2dh=ardr  ......     (5) 

If  b  is  a  constant  of  integration,  eq.  (5)  may  be  integrated 
so  as  to  take  the  form  of  eq.  (6)  : 

r3h=-ar3+b-     h=-+^  .....     (6) 
3  3     r3 

By  using  the  first  of  eqs.  (4)  and  eq.  (6),  eq.  (7)  at  once 
follows  : 

a     2b  x 


At  the  inner  and  outer  surface  of  the  spherical  shell 
p  =p'  and  p  =  pi,  respectively.     Eq.  (7)  will  then  give: 


,     a     2b       -          a     2b  /ox 

=---       •'       •       •       •        (8) 


Hence  : 


ri 


The  preceding  equation  will  at  once  give  the  following 
value  of  b,  which  in  turn  substituted  in  the  second  of  eqs. 

8  will  give  the  value  of  -,  following  that  of  b: 


228  HOLLOW  CYLINDERS  AND  SPHERES.  [Ch.  IV. 

These  values  of  b  and  —  substituted  in  eqs.  (6)  and  (7) 

o 

will  give  the  following  values  of  radial  intensity  p  and  cir- 
cumferential intensity  h  at  any  point  in  the  shell  distant 
r  from  the  centre.  In  writing  these  final  expressions  it  is 
to  be  remembered  that  the  constants  a  and  b  may  be  either 
positive  or  negative  and  their  signs  are  changed  so  as  to 
make  all  positive  stress  tension  and  all  negative  stress  com- 
pression, as  was  done  in  the  case  of  thick  cylinders. 

' 


(r'3-n3) 


,       .  , 

'*"  —  To  Q  '  /    /Q  o\          o  •  •         •         \*-  *-  J 

r'3—n3  2(r3—ri3)    r3 

These  equations  can  be  put  in  more  convenient  shape 
for  computation  by  dividing  all  terms  in  the  second  mem- 
bers by  ri3,  which  will  give  eqs.  (ioa)  and  (na)  : 

'3 


,r/3 

^-^T3       (^,^3, 

r'3  /3  ^3'         '        '        '        ^I] 

—  r-I  —  T-I 

n3  fi3 

It  is  necessary  to  determine  a  thickness  /  of  shell  which 
will  resist  a  given  intensity  of  internal  pressure.  Eq.  (n) 
shows  that  the  circumferential  tension  h  will  be  greatest 
when  r=rf  in  eq.  (n).  Making  this  substitution 


Art.  42.]  THICK  HOLLOW  SPHERES.  229 

Dividing  by  r'3  and  solving: 

n3       2(h+p') 


Hence  there  may  be  at  once  written  : 


This  value  of  t  will  give  the  thickness  of  material  re- 
quired so  that  the  maximum  intensity  of  circumferential 
tensile  stress  shall  not  exceed  a  prescribed  value  h  at  the 
interior  surface  of  the  sphere  when  the  interior  pressure  is 
p'  and  the  exterior  pressure  pi,  the  latter  being  smaller 
than  the  former. 

A  similar  treatment  may  be  given  to  eq.  (n)  after 
making  r—r\  in  order  to  determine  a  thickness  t  such  that 
the  circumferential  compressive  stress  .shall  not  exceed  a 
given  prescribed  value  when  the  exterior  pressure  pi  exceeds 
the  interior  pressure  p' '. 

In  eq.  (12)  if  p'  =  2h-\-$pi,  the  value  of  t  becomes 
infinitely  great,  showing  that  if  the  interior  pressure  reaches 
or  exceeds  the  value  indicated  no  thickness  of  shell  what- 
ever will  prevent  the  circumferential  or  hoop  tension  ex- 
ceeding the  prescribed  limit  h. 

If  either  internal  or  external  pressure  become  zero, 
while  the  other  has  any  assigned  value,  it  is  only  necessary 
to  make  either  p'  or  pi  equal  zero  in  all  the  preceding 
equations.  Furthermore,  it  is  a  matter  of  indifference 
whether  p1  or  pi  is  numerically  greater  in  the  application 
of  any  of  the  preceding  equations  except  eq.  (12). 


230  HOLLOW  CYLINDERS  AND  SPHERES.  [Ch.  IV. 

Radial  Displacement  at  any  Point  in  the  Spherical  Shell 

The  general  analysis  of  Art.  8  of  Appendix  I,  gives  an 
expression  for  the  radial  strain  or  displacement  of  the 
material  at  any  point  within  the  spherical  shell.  It  has 
already  been  seen  that  no  other  displacement  occurs,  as 
all  diametral  sections  of  the  shell  remain  plane  for  any 
degree  of  stress  whatever.  If  this  radial  displacement  or 
strain  at  any  point  be  indicated  by  p,  the  analysis  indi- 
cated shows  that  the  value  of  this  displacement  will  be 
given  by  eq.  (13)  : 

r'3 

Pl~P'^  ,  (^-pya/aj 
"  ~~      ~  ' 


—  I 


Knowing  the  internal  and  external  pressures  to  which 
the  shell  is  subjected  eq.  (13)  will  give  the  value  of  the 
radial  displacement  of  any  indefinitely  small  piece  of 
material  at  the  distance  r  from  the  center.  If  r  =  r\  the 
corresponding  value  of  p  given  by  eq.  (13)  will  indicate 
the  increase  or  decrease,  as  the  case  may  be,  of  the  external 
radius  r\\  and  if  r  =  rr  the  increase  or  decrease  in  length 
of  the  interior  radius  r'  will  result.  In  eq.  (13)  G  is  ob- 
viously the  modulus  of  elasticity  of  the  material  for  shear, 
while  r  is  the  ratio  of  lateral  over  direct  strains. 


CHAPTER  V. 

RESILIENCE. 

Art.  43. — General  Considerations. 

THE  term  resilience  is  applied  to  the  quantity  of  work 
required  to  be  expended  in  order  to  produce  a  given  state 
of  strain  in  a  body.  If  a  piece  of  material  is  subjected 
to  tension,  that  state  of  strain  will  be  simply  the  stretching 
of  the  piece  or  the  amount  of  compression,  if  the  piece  is 
subjected  to  compressive  stress.  In  precisely  the  same 
manner  the  resilience  of  a  bent  beam  is  the  amount  of  work 
performed  upon  it  by  its  load  in  producing  deflection. 
There  may  also  be  the  resilience  of  shearing  or  of  torsion. 

In  the  ordinary  use  of  the  expression,  resilience  refers 
to  the  amount  of  work  expended  within  the  elastic  limit, 
whether  of  torsion,  compression,  or  tension,  but  it  may 
properly  be  extended  in  its  meaning  to  include  the  total 
amount  of  work  required  to  rupture  the  material  under 
any  one  of  the  preceding  conditions  of  stress.  Elastic 
resilience  may  easily  be  computed  by  means  of  exact 
formulae,  but  if  the  total  work  required  to  cause  rupture 
in  any  case  is  desired,  a  graphical  record  of  the  total  strains 
produced  between  the  elastic  limit  and  failure  must  be 
obtained  by  actual  tests.  In  these  articles  the  formulae 
for  elastic  resilience  only  will  be  given;  in  other  subse- 
quent articles  the  method  of  computing  the  total  resilience 

231 


232  RESILIENCE.  [Ch.  V. 

of  failure  will  be  illustrated  by  computations  from  actual 
strain  records. 


Art.  44.  —  The  Elastic  Resilience  of  Tension  and  Compression 
and  of  Flexure. 

Let  it  be  supposed  that  a  piece  of  material  whose  length 
is  L  and  the  area  of  whose  cross-section  is  A  is  either 
stretched  or  compressed  by  the  weight  or  load  W  applied 
so  as  to  increase  gradually  from  zero  to  its  full  -value.  If 
E  is  the  coefficient  of  elasticity,  the  elastic  change  of  length 

WL 

will  be  ~A~'     The  average  force  acting  will  be  %W,  hence 


the  work  performed  in  producing  the  strain  will  be 


W2L 
Resilience  =  —  j-    .......     (i) 


W 
If  -j-,  the  intensity  of  stress  in  the  metal,  be  represented 

by  t,  eq.  (i)  may  be  written 

Resilience  =%At2-^ (2) 

Again,  eq.  (2)  may  take  the  following  form: 

t2 
Resilience  =  ^AE-^-2L  =  %AE12L.      .     ...     (3) 


t2 
In  eq.   (3)   the  quantity  l2=^r-2  is  obviously  the  square 

of  the  strain  (stretch  or  compression)  per  unit  of  length. 
If  a  bar  of  material  i  inch  in  length  and  i  square  inch 


Art.  44.]  RESILIENCE  OF  BENDING  OR  FLEXURE.  233 

in  cross-section  be   considered,  -A  =  i   and  L  =  i  must  be 
inserted  in  the  preceding  equations,  and  there  will  result 

t2 
Unit  resilience  =  %p=%El2.      .....     (4) 

t2 

The  quantity  —  =EP  is  called  the  "  Modulus  of  Resil- 
ience." The  expression  is  ordinarily  employed  when  t 
is  the  greatest  intensity  of  stress  allowed  in  the  bar. 

The  preceding  equations  are  applicable  whether  the 
bar  or  piece  of  material  is  in  tension  or  compression,  the 
coefficient  of  elasticity  E  being  used  for  either  stress,  while 
/  represents  the  intensity  of  either  tension  or  compression, 
as  the  case  may  be. 

Inasmuch  as  the  values  of  t  and  E  are  usually  taken 
in  reference  to  the  square  inch  as  the  unit  of  area,  it  is 
generally  convenient  to  take  L  in  inches,  although  any 
other  unit  of  length  may  be  taken  when  multiplied  by  the 
proper  numerical  coefficient. 


The  Resilience  of  Bending  or  Flexure. 

It  has  already  been  shown,  in  considering  the  common 
theory  of  flexure,  as  applied  to  the  flexure,  or  bending  of 
beams,  that  the  intensities  of  the  stresses  of  tension  and 
compression  vary  from  point  to  point  throughout  the 
entire  beam.  In  determining  the  elastic  resilience  of 
flexure,  therefore,  it  is  necessary  to  find  the  work  per- 
formed in  producing  the  varying  strains  corresponding 
to  the  stresses  in  the  interior  of  the  beam.  The  resilience 
due  to  the  direct  stresses  of  tension  and  compression  will 
first  be  considered  and  then  that  due  to  the  shearing 
stresses. 


234  RESILIENCE.  [Ch.  V. 

In  order  to  obtain  the  expression  for  the  work  per- 
formed by  the  direct  stresses  of  tension  and  compression 
in  a  beam  bent  by  loads  acting  at  right  angles  to  its  axis, 
a  differential  of  the  length,  dLt  is  to  be  considered  at  any 
normal  section  in  which  the  bending  moment  is-M,  the 
total  length  of  span  or  beam  being  L.  Let  /  be  the  moment 
of  inertia  of  the  normal  section,  A,  about  its  neutral  axis, 
and  let  k  be  the  intensity  of  stress  (usually  the  stress  per 
square  inch)  at  any  point  distant  d  from  the  axis  about 
which  /  is  taken.  The  elastic  change  produced  in  the 
indefinitely  short  length  dL  when  the  intensity  k  exists 

k 
is  -pdL.     If  dA  is  an  indefinitely  small  portion  of  the  normal 

section,  the  average  force  or  stress,  either  of  tension  or 
compression,  acting  through  the  small  elastic  change  of 
length  just  given,  can  be  written  by  the  aid  of  a  familiar 
equation  of  flexure  as 

A  .......     (5) 

Hence  the  work  performed  in  any  normal  section  of  the 
member,  for  which  M  remains  unchanged,  will  be,  since 

fk.dA.d=M, 

M  M2 


(6) 


The  work  performed  throughout  the  entire  piece  will  then 
be 


The  integration   indicated   in  eq.    (7)  is  readily  made 
in  all  ordinary  cases  by  substituting  the  value  of  the  bend- 


Art.  44.]  RESILIENCE  OF  BENDING   OR  FLEXURE.  235 

ing  moment  M  in  terms  of  the  variable  horizontal  ordinate 
or  abscissa  x  and  the  load,  it  being  remembered  that  dL 
is  precisely  the  same  as  doc.  If,  for  example,  the  beam 
is  non-continuous,  simply  supported  at  each  end  and 
carries  uniformly  distributed  load  p  per  unit  of  length 

P 

throughout  the  whole   span,   M=-(Lx  —  x2).     By  the  in- 

sertion of  this  value  of  M  in  eq.  (7),  there  will  result 

j       rL  p2xz  p-ijj        W2L3 

Resilience  =  —^  I     ——(L  —  x)2dx  =        r.T=  -  F^,     (8) 
2EIJo       4  24oEI     24oEP 


W  representing  pL,  the  entire  load  on  the  beam. 

This  equation  gives  the  value  of  the  total  work  per- 
formed by  the  direct  stresses  of  tension  and  compression 
in  the  interior  of  a  simple  .  beam  uniformly  loaded  and 
supported  at  each  end,  under  the  assumption  that  the 
moment  of  inertia  7  of  the  cross-section  is  constant  through- 
out the  entire  span. 

If  a  single  load  W  rests  at  the  centre  of  the  span,  the 

W 
reaction  at  each  end  being  —  ,  the  value  of  the  bending 

W 
moment  at  any  point  will  be  —  x.     By  inserting  this  value 

of  M  in  eq.  (7),  there  will  result 

i          n  W2  2  ;       W2L* 

Resilience  =—  ^.2  /      —  x*dx  =  .     .     (9) 

2EI     Jo      4  g6EI 

Similarly  equations  of  the  elastic  resilience  of  the  direct 
stress  of  tension  and  compression  in  beams  loaded  in  any 
manner  whatever  may  easily  be  written.  In  some  cases 
like  the  last  the  deflection  at  the  point  of  application  of  a 
single  load  may  easily  be  determined.  Let  that  deflec- 


236  RESILIENCE.  [Ch.  V. 

tion  be  represented  by  w\  when  a  single  load  W  rests  at 
the  centre  of  the  span  the  work  performed  by  this  load  in 
producing  the  deflection  is  %Ww.  Hence  that  amount  of 
work  must  be  equal  to  the  resilience  given  by  eq.  (9),  and 

WL3 

•••••••     do) 


The  Resilience  Due  to  the  Vertical  or  Transverse  Shearing 
Stresses  in  a  Bent  Beam. 

The  work  performed  by  the  vertical  shearing  stresses 
in  a  bent  beam  of  any  shape  of  cross-section  may  readily 
be  found.  Let  5,  be  the  total  transverse  shear  in  a  normal 
section,  /  being  the  moment  of.  inertia  of  the  latter  about 
its  neutral  axis,  b  the  width  or.  breadth  (constant  or  variable) 
of  the  section,  <f>  the  unit  shearing  strain  denned  in  Art.  2, 
d  and  dl  the  distances  of  the  extreme  fibres  from  the  neu- 
tral axis,  and  G  the  coefficient  of  elasticity  for  shearing. 
By  eq.  (6)  of  Art.  15  the  intensity  5  of  the  shear  at  any 
point  in  the  section  at  the  distance  z  from  the  neutral  axis 
will  be 

s=~(d*-z*)  .......     (u) 

Again,  by  eq.  (3)  of  Art.  2, 


The  amount  of  shearing  stress  on  the  indefinitely  small 
portion  of  the  section  b.dz  will  be  sb.dz,  and  its  path  in 
performing  the  work  will  be  <£>dx,  x  being  the  horizontal 
ordinate  of  the  section  of  the  beam  from  any  convenient 
origin,  as  the  end  or  the  centre  of  the  span,  i.e.,  in  this  case 


Art.  44-1  RESILIENCE  DUE    TO  SHEARING  STRESSES.  237 

• 

the  end  of  the  span.     The  differential  work  performed  in 
the  section  will  be,  by  the  aid  of  eqs.  (n)  and  (12), 


.     .     (13) 


In  this  equation  it  is  easy  to  express  the  breadth  b  of 
the  section  in  terms  of  z,  whatever  may  be  its  shape,  by 
the  aid  of  the  equation  of  the  perimeter  of  the  section. 
In  all  the  ordinary  and  important  cases  of  engineering 
practice  involving  this  resilience  of  shearing  the  shape  of 
the  section  is  rectangular  for  which  b  is  constant,  and  it 
will  be  so  regarded  in  the  following  equations.  Remem- 
bering that  x  and  z  are  independent  variables,  and  that 
the  first  integration  will  be  made  in  reference  to  z,  that 
integration  will  give 


As  the  section  is  taken  to  be  rectangular  in  outline,  with 
the  breadth  b  and  depth  h,  d=d1=-  and  /  = — .     Eq.  (14) 

will  then  become 

3       /» 

Resilience  =-j-j-^  I  S2dx.    .     .     .     .     (15) 


The  total  transverse  shear  5  will  have  varying  values 
depending  upon  the  amount  of  loading  on  the  beam  and 
its  distribution,  i.e.,  in  general  it  will  vary  with  .x,  and 
when  not  constant  it  must  be  expressed  in  terms  of  that 
variable  before  the  remaining  integration  can  be  made. 

If  a  single  weight  W  rests  on  the  beam  at  the  distance 
of  /'  from  one  end  where  the  reaction  is  R',  and  at  the 


238  RESILIENCE.  [Ch.  VI. 

distance  /t  from  the  other  end  where  the  reaction  is  Rv 
the  shear  5  will  be  constant  for  each  of  the  segments  into 
which  the  point  of  loading  divides  the  span  ;  in  one  of 
those  segments  S  =  R',  and  in  the  other  S  =  Rr  The  com- 
plete integration  of  eq.  (15)  will  be,  therefore, 

Resilience  -  -(Rr2lf  +Rl2ll). 


If  there  be  substituted  in  the  parentheses  of  the  second 

member  of  the    preceding   equation  the   values  Rf  =  Wj 

lr 
and  Ri  =  Wj,  there  will  result 

3      IV 

Resilience  =—  r.--^  ^j-W2.     .     .     ;     .     (16) 
$bhG    I 

If  the  weight  W  rest  at  the  centre  of  the  span  /t  =  V  =  - 


and 


Resilience  =—~jWH (17) 

2oGbh 


Eq.  (17)  affords  a  simple  method  of  finding  the  deflec- 
tion w1  of  the  point  of  loading  due  to  the  transverse  shear. 
As  the  weight  W  is  supposed  to  be  gradually  applied  the 
expended  work  ^Wwl  must  be  equal  to  the  shearing  re- 
silience given  in  eq.  (17).  Hence 

Wi=^~  -rr.     .      .      .  (18) 

loG    oh 

When  a  non-continuous  beam  simply  supported  at  each 
end  carries  a  uniform  load  over  the  entire  span,  it  has  been 
shown  in  Art.  22,  eq.  (7),  that  the  transverse  shear  at  any 


Art.  44.].  DIRECT  AND  SHEARING  STRESSES.  239 

section  is  equal  to  the  load  between  the  centre  of  span  and 
that  section.  If,  therefore,  the  origin  of  x  be  taken  at 
the  centre  of  span  and  if  p  represents  the  load  per  unit  of 
length  of  the  beam,  S=px.  By  substituting  this  value  of 
5  in  eq.  (15),  and  remembering  that  twice  the  integral 
must  be  taken  for  the  whole  beam, 

i 

Resilience  =  -^rr  .  2  I     pVdx  =    ^       =  —7^7 .    (19) 
$Gbk     Jo  2oGbh     2oGbh 

The  shearing  resilience,  therefore,  in  a  non-continuous 
beam  carrying  a  uniform  load  is  only  one  third  as  much 
as  that  due  to  the  same  load  concentrated  at  the  centre  of 
the  span. 

If,  as  is  usual,  G  is  expressed  in  pounds  per  square  inch 
the  unit  for  I,  b,  and  h  will  be  the  linear  inch. 

Other  modes  of  loading  than  those  taken  can  be  treated 
in  precisely  the  same  general  manner. 

As  the  intensity  of  the  longitudinal  shear  at  any  point 
of  a  beam  is  the  same  as  that  of  the  transverse  shear, 
the  total  work  of  the  longitudinal  shear  throughout 
the  beam  is  the  same  as  the  work  of  the  transverse 
shear.  The  total  work  of  the  shearing  stresses  in  a  beam 
is  therefore  composed  of  those  two  equal  parts. 

The  Total  Resilience  Due  to  Both  Direct  and  Shearing 

Stresses. 

The  general  expression  for  the  total  resilience  of  a  bent 
beam  due  to  both  shearing  and  direct  stresses  will  be  the 
sum  of  the  second  members  of  eqs.  (7)  and  (13),  expressed 
by  the  following  equation: 

/M 2  C   C  S2b 

^£jdL  +  J  J 


240      .  RESILIENCE.  [Ch.  V. 

Or,  by  eqs.  (7)  and  (15),  since  dL=dxy 

/M2  *       C 

-^jdx  +  -TT-F.  /  S2dx.     .      (20) 

By  the  aid  of  eqs.  (8)  and  (19)  the  total  resilience  for 
a  simple  non  -continuous  beam  may  be  as  follows: 
If  the  uniform  load  pi  =  W, 


Total  resilience—  W*[  --  ~H  —  7^7  )  .       .     (21) 
\  24oEI     2oGbh/ 

For  the  same  beam  carrying  a  single  load  W  at  the  centre, 
by  eqs.  (9)  and  (17) 

Total  resilience  =  W2\    .^T  +     *•  )  .      .     .     (22) 
\()6EI     2oGbh/ 

As  has  been  explained,  the  last  two  equations  are  appli- 
cable to  beams  with  rectangular  sections  only. 

In  a  similar  manner  the  total  deflection  of  a  beam 
supported  at  each  end  and  loaded  with  a  single  weight  W 
at  the  centre  of  the  span,  due  to  bending  and  flexure,  will 
be  found  by  the  sum  of  the  two  expressions  given  in  eqs. 
(10)  and  (18): 


Art.  45. — Resilience  of  Torsion. 

The  work  expended  in  producing  elastic  strains  of 
torsion  constitutes  the  resilience  of  torsion  and  is  a  special 
case  of  shearing  resilience.  The  twisting  moment  which 
produces  the  angle  of  torsion  a  is  given  by  eq.  (16)  of 


Art.  44-1  RESILIENCE  OF  TORSION.  241 

Art.  37  and  is  M  =  GaIp.  When  the  piece  twisted  has  the 
length  /  the  total  angle  of  torsion  is  al  and  the  differential 
amount  of  work  performed  by  the  moment  M  in  producing 
the  indefinitely  small  twist  d(al)  =l.da  is  Ml.  da.  Hence 

Resilience  -  f  Mlda  =  GIIP  f^a  .  da  =  Gllp°^-  .        (  i  ) 

«/  «/  0  2 

If  P  and  e  are  the  force  and  lever-arm  of  the  twisting 
couple,  eq.  (18)  of  Art.  37  shows  that 

Pe 


Substituting  this  value  of  al  in  eq.  (i), 

P2e2L 
Resilience  =  —-  .  .     .     .     .     .     .     (2) 


nr4 
If  the  normal  section  of  the  piece  is  circular  Ip  =  --- 

Hence,  for  a  shaft  with  circular  section, 

Pze*l 


Resilience 


64 
If  the  section  of  the  shaft  is  a  square,  IP  =—,  b  being 

the  side  of  the  square.     Hence,  for  a  square  section, 

Resilience  =  — -^rr  •  (4) 

Go* 

In  some  cases  shafts  are  subjected  to  combined  torsion 
and  bending.  In  such  cases,  if  it  is  desired  to  compute 
the  total  elastic  resilience  it  is  only  necessary  to  take  the 


242  RESILIENCE.  [Ch.  V. 

stun  of  the  two  resiliences,  each  found  as  if  existing  in- 
dependently  of  the  other. 

The  resilience  of  torsion  beyond  the  elastic  limit  or 
between  the  elastic  limit  and  the  ultimate  resistance  must 
be  determined,  as  in  all  cases  of  distortion  beyond  the 
elastic  limit,  from  an  actual  strain  record,  as  given  by 
the  testing  machine  when  the  piece  is  strained  up  to  any 
given  degree  of  permanent  stretch  or  to  rupture . 

Art.  46. — Suddenly  Applied  Loads. 

A  load  is  considered  suddenly  applied  when  its  full 
amount  acts  instantly  upon  any  piece  of  material  loaded 
by  it.  In  the  preceding  articles  relating  to  resilience  the 
loads  are  treated  as  being  gradually  increased  from  zero 
to  their  full  values.  In  such  cases  the  amount  of  external 
loading  at  any  instant  is  supposed  to  be  equal  only  to  the 
internal  stress  or  stresses  opposing  it,  so  that  the  work 
performed  is  equivalent  to  one  half  the  total  load  multi- 
plied by  the  total  resulting  strain.  When  the  loads  are 
suddenly  applied,  on  the  other  hand,  the  internal  stresses 
produced  are  exactly  equal  to  the  external  forces  only 
when  the  strains  corresponding  to  the  latter  are  reached, 
and  the  work  performed  up  to  that  point  is  just  double 
the  work  expended  when  the  loads  are  gradually  applied. 
It  follows  from  this  last  consideration  that  the  strains 
produced  by  the  suddenly  applied  loads  will  be  double 
those  found  under  gradual  application.  Inasmuch  as 
the  elastic  strains  are  proportional  to  the  corresponding 
stresses,  it  further  follows  that  the  stresses  produced  by 
suddenly  applied  loads  will  be  double  in  intensity  those 
which  are  produced  by  the  same  loads  gradually  applied. 

The  work  expended  by  a  suddenly  applied  load  up  to 
the  point  of  strain  corresponding  to  its  amount  being 


Art.  46.]  PROBLEMS  FOR   CHAPTER  V.  243 

double  the  work  performed  by  the  internal  stresses,  the 
total  stress  induced  in  the  material  at  the  limit  of  the  final 
strain  produced  by  such  a  load  will  be  double  the  amount 
of  the  latter.  The  internal  stresses  in  the  piece  will,  there- 
fore, cause  it  to  recover  from  its  strained  condition  and 
vibrations  will  result,  the  treatment  of  which  constitutes 
an  important  branch  of  the  theory  of  elasticity  in  solid 
bodies.  Some  general  features  of  that  treatment  will  be 
given  in  Art.  12,  App.  I,  but  as  they  are  seldom  used  in 
engineering  practice  they  will  not  be  considered  here. 
It  is  only  important  at  this  point  to  note  carefully 
the  distinction  between  the  effects  of  a  given  load  grad- 
ually applied  and  suddenly  applied,  the  strains  and 
stresses  in  the  latter  condition  being  double  those  in 
the  former. 

Again,  it  is  also  important  to  distinguish  between 
loads  suddenly  applied,  and  shocks,  as  they  are  called  in 
engineering  practice.  A  shock  is  produced  when  the 
load  falls  freely  before  acting  upon  a  piece  of  material 
sustaining  it.  The  cause  of  shock,  therefore,  is  a  suddenly 
applied  load  with  the  effect  of  a  free  fall  of  the  latter  super- 
imposed. These  matters  must  be  carefully  taken  into 
account  and  allowed  for  in  such  structures  as  bridges 
carrying  rapidly  moving  trains,,  and  those  allowances  are 
incorporated  in  the  provisions  of  specifications  covering 
bridge  construction. 

PROBLEMS  FOR  CHAPTER  V. 

• 

Problem  i. — A  6-inch  by  i. 75-inch  steel  eye-bar  48  feet 
long  is  subjected  to  a  stress  of  117,500  pounds.  If  that 
load  is  gradually  applied  what  is  the  work  performed  in 
the  total  length  of  the  bar,  if  E  =  30,000,000  pounds?  Also 
what  is  the  unit  resilience? 


244  RESILIENCE.  •  [Ch.  V. 

t  =  — — — -=11,190.     L  =  48 X  12  =576  inches.     Eq.    (2) 
10.5 

of  Art.  44  then  gives 
Resilience  =work  performed  = 


2X30,000,000 
=  12,621  in.-lbs. 
Eq.  (4)  of  Art.  44  gives 


.,.  .  . 

Unit  resilience  =  —  -  =2.09  m.-lbs. 

2  X  30,  000,000 

Problem  2.  —  A  cast-iron  column  18  feet  long  having 
an  area  of  cross-section  of  40.8  sq.  in.  carries  a  load  of 
245,000  pounds.  If  the  coefficient  of  elasticity  E  is  14,- 
000,000  pounds,  how  much  work  is  performed  in  com- 
pressing the  column  if  the  load  is  gradually  applied. 

Problem  3.  —  A  3o-pound  lo-inch  rolled  steel  I  beam 
carries  a  uniform  load  of  1000  pounds  per  linear  foot  in 
addition  to  its  own  weight  with  a  span  of  16  feet.  What 
will  be  the  resilience  or  work  performed  in  the  material 
of  the  beam  under  the  gradual  application  of  that  total 
load  of  1030  pounds  per  linear  foot,  the  moment  of  inertia 
/  of  the  beam  being  134.2  and  £  =  30,000,000  pounds? 
Eq.  (8)  of  Art.  44  is  to  be  used,  in  which  L  is  192  inches. 
Incidentally,  what  will  be  the  greatest  intensity  of  stress, 
k,  in  the  extreme  fibres? 

Ans.  Resilience  =  1987  in.-lbs  ;  k  =  15,000  Ibs.  per  square 
inch. 

Problem  4.  —  In  Problem  3  if  the  thickness  of  the  web 
of  the  lo-inch  rolled  beam  is  .5  inch,  find  the  resilience  of 
the  vertical  or  transverse  shearing  stresses  in  the  beam, 
the  coefficient  of  shearing  elasticity,  G,  being  taken  at 
12,000,000  pounds.  The  remaining  data  are  1  =  192  inches; 
h  =  io  inches;  6=0.5  inch,  and  1/^  =  16,480  pounds,  and 
they  are  to  be  used  in  eq.  (19)  of  Art.  44. 


Art.  46.]  PROBLEMS  FOR  CHAPTER   V.  245 

Problem  5. — A  round  bar  of  steel  2f  inches  in  diameter 
is  twisted  by  a  force  of  2100  pounds  acting  with  a  lever- 
arm  of  17  inches.  Two  sections  25  ft.  apart  are  turned 
0.185  inch  in  reference  to  each  other,  i.e.,  the  total  strain 
of  torsion  for  a  length  of  bar  of  25  feet  has  that  value. 
Find  the  total  angle  of  torsion,  the  angle  of  torsion  and  the 
coefficient  of  elasticity,  G,  for  shearing  (i.e.,  for  torsion). 
Ans.  01=0.00043;  a/=o.i2g;  and  G  =  13,000,000  Ibs. 

Problem  6. — The  greatest  permitted  working  intensity 
of  torsive  shearing  is  8000  pounds  per  square  inch.  Design 
a  steel  shaft  to  carry  a  twisting  moment  produced  by  a 
force  of  1900  pounds,  acting  with  a  lever-arm  of  84  inches. 
If  the  coefficient  of  elasticity  for  shearing  is  12,000,000 
pounds,  what  will  be  the  angle  of  torsion?  Also  what  will 
be  the  total  angle  of  torsion  and  total  strain  of  torsion  for 
a  length  of  shaft  of  13  feet? 

Problem  7. — In  Problems  5  and  6  find  the  work  per- 
formed in  twisting  the  two  steel  shafts,  i.e.,  the  resilience 
for  25  feet  length  in  the  one  case  and  13  feet  in  the  other. 
Use  equations  of  Art.  45. 

Problem  8. — In  Problem  5  suppose  the  load  suddenly 
applied,  what  will  be  the  resulting  resilience  and  greatest 
intensity  of  extreme  fibre  stress? 


CHAPTER    VI. 
COMBINED   STRESS   CONDITIONS. 

Art.  47. — Combined  Bending  and  Torsion. 

PROBABLY  the  most  important  case  of  combined  bend- 
ing or  flexure  and  torsion  is  that  of  the  ordinary  crank- 
shaft represented  in  Fig.  i. 

The  centre  of  the  thrust  of  a  connecting-rod  is  at  A, 
on  the  crank-pin  journal  against  which  the  connecting-rod 
bears.  The  centre  of  the  shaft-bearing  is  at  B.  If  the 
thrust  at  A  is  represented  by  P,  then  the  actual  resultant 
moment  about  the  centre  of  the  bearing  B  will  be  PxAB. 
The  problem  is  to  determine  the  maximum  stresses  de- 
veloped by  this  resultant  moment  in  the  section  of  the 
shaft  at  B.  Two  methods  may  be  employed  in  both  of 
which  the  resultant  moment  of  P  multiplied  by  the  lever 
arm  A  B  is  resolved  into  its  two  components,  one  of  which 
is  the  ordinary  bending  moment  represented  by  M  =  P  X  CB, 
and  the  other  is  the  twisting  moment  M'  =  PxAC.  The 
la.tter  produces  torsion  in  the  journal  at  B  and  the  former 
produces  pure  flexure  or  bending  at  the  same  section. 

Let  CB  be  represented  by  /  while  e  represents  AC. 
The  moment  of  pure  bending  at  B  will  be 

M=Pl. (i) 

246 


Art.  47.]  COMBINED  BENDING  AND    TORSION.  .     247 

The  twisting  moment  producing  pure  torsion  will  be 

M'=Pe.     ,\, (2) 

If  d  represents  the  distance  of  the  most  remote  fibre 
in  the  section  B  from  the  neutral  axis  of  the  latter,  and  if 


FIG.   i. 

k  is  the  greatest  intensity  of  bending  stress  at  the  dis- 
tance d  from  the  neutral  axis,  while  /  is  the  moment  of 
inertia  of  the  normal  section  of  the  shaft  at  B  about  the 
same  neutral  axis,  the  following  will  be  the  value  ot  k : 

Md    Pld 


(3) 


Again,  if  T  is  the  greatest  intensity  of  torsional  shear 
in  the  normal  section  of  the  shaft  at  B,  at  the  greatest 
distance  r,  in  the  perimeter,  from  the  centre  of  gravity  or 
the  centroid  of  the  same  section,  the  value  of  the  maximum 
intensity  T  will  be 


Mfr_Per 


(4) 


In  eq.  (4)  Ip  is  the  polar  moment  of  inertia  of  the  nor- 
mal section  at  B. 


248  COMBINED  STRESS   CONDITIONS.  [Ch.  VI. 

First  Method. 

In  this  method  it  is  only  necessary  to  consider  the 
intensities  k  given  by  eq.  (3)  and  T  given  by  eq.  (4),  the 
greatest  allowed  working  stresses  of  direct  tension  and  of 
shearing  respectively,  k  would  have  the  value  of  the 
greatest  tensile  working  stress  of  the  material  of  the  shaft 
for  the  reason  that  if  tested  to  failure  the  shaft  would 
yield  first  on  the  tension  side. 

It  being  understood,  therefore,  that  k  and  T  represent 
the  greatest  allowed  working  intensities  of  stress,  usually 
expressed  in  pounds  per  square  inch,  eq.  (3)  will  give 

I     M     PI 

5=T=T (5) 


Under  the  same  conditions  eq.  (4)  will  give 

Pe 


I,     M'     Pe 


For  the  circular  section 


7~— <     and    4=^-.       ....     (7) 
For  a  square  section 

/-         and    /,-, (8) 


b  being  the  side  of  the  square.     In  eq.  (5)  for  a  circular 
section  d  =  r  and  for  a  square  section  d  =  — -=.     In  eq.  (6), 

V  2 

r  =  r  for  the  circular  section,  but  for  the  square  section 


Art.  47.]  COMBINED  BENDING  AND   TORSION.  249 

r«-7=.     Making  those  substitutions  in  eqs.  (5)  and  (6)  for 

V  2 

the  circular  section,  there  will  result,  D  being  the  diameter 
of  the  shaft, 


„     ,       ,.  D     ,I4W  s  P/] 

tor  bending  .  .  .  r  = —  =-v  nr  =  i.oo-v^-r- 

2  il    /;./?  *    A? 

•        (9) 


For  torsion  . .  . .  r  =—  r=\j-*-T=r  =  .po\i-~* 

2  ^    7T7  ^    J! 

In  the  practical  use  of  eqs.  (9)  that  one  of  the  two 
values  of  r  should  be  taken  which  is  the  greatest.  This 
will  insure  that  both  the  direct  stress  of  tension  and  the 
shearing  stress  shall  not  exceed  the  prescribed  values  of 
k  and  T. 

The  substitution  of  the  values  of  /  and  IP  for  the  square 
section  in  eqs.  (5)  and  (6)  will  give,  remembering  that  d 
and  r  are  each  one  half  the  diagonal  of  the  square, 


t*     t;     ^  L  Pl 

For  bending  .  .  .  b  =\ 


3 
For  torsion  .  .  .  .  b  =  z.6?\s; 


.      do) 


In  eq.  (10),  also,  the  greatest  value  of  b  given  by  the 
application  of  the  two  formulae  is  to  be  taken,  so  that,  as 
in  the  case  of  the  circular  section,  neither  of  the  two  in- 
tensities k  and  T  shall  exceed  the  values  prescribed  for 
them. 

This  method  involves  only  the  consideration  of  the 
simple  formulae  of  the  common  theories  of  flexure  and 
torsion. 


2$0  COMBINED  STRESS  CONDITIONS.  [Ch.  VI. 

Second  Method. 

The  second  method  of  treatment   of  this  case  of  the 
crank-shaft  consists  in  determining  the  greatest  intensity 

of  the  direct  stress  of  tension  in  the  section  B  of  the  shaft 

« 

at  the  journal-bearing.  This  resultant  maximum  intensity 
is  produced  by  the  combination  of  the  same  component 
moments,  M  =Pl  and  Mf  =Pe,  as  in  the  preceding  method. 
With  the  sections  of  shafting  always  employed  the  maxi- 
mum intensity  of  bending  stress  k  and  the  maximum 
intensity  of  torsional  shear  T  exist  at  the  same  point  and 
on  the  same  plane,  i.e.,  the  plane  of  normal  section  of  the 
shaft.  The  existence  of  the  shear  T  on  the  normal  section 
at  the  distance  r  from  its  centre  of  gravity  carries  with  it 
the  same  intensity  of  shear  at  the  same  point  on  a  longi- 
tudinal plane  passing  through  the  axis  of  the  shafting. 
At  the  .  point  considered,  therefore,  on  two  indefinitely 
small  planes  at  right  angles  to  each  other,  one  normal  to 
the  axis  of  the  shaft  and  the  other  parallel  to  it,  there  exist 
the  direct  intensity  of  tension  k  on  the  first,  and  the 
intensity  of  shear  T  on  the  second.  The  problem  is  to 
determine  at  the  same  point  the  greatest  intensity  of 
the  direct  stress  of  tension  on  any  plane  whatever,  and 
the  angle  ft  between  the  direction  of  that  stress  and  the 
axis  of  the  shaft.  Reference  may  best  be  made  to  the 
general  formulae  of  internal  stresses  in  a  solid  body  for  its 
solution,  and  those  are  eqs.  (8)  and  (9)  of  Art.  8.  Those 
equations  are  adapted  to  this  case  by  making  px=k, 
pXy  —  T,  tan  «=tan  /3,  and  p  =  t,  the  latter  quantity  being 
the  greatest  intensity  of  tension  desired.  These  substi- 
tutions give  the  following  two  equations : 

k 

* („) 

2 


Art.  47.]  COMBINED   BENDING   AND   TORSION.  251 

T 

tan  2|8=--r- (12) 

K 


Eq.  (n)  gives  the  greatest  intensity  of  direct  tension 
in  the  shaft  in  terms  of  known  stresses. 

By  eq.  (12)  the  position  of  the  plane  or  section  of  the 
shaft  on  which  the  maximum  intensity  t  exists  may  at 
once  be  found.  Inasmuch  as  /3  is  the  angle  between  the 
direction  of  the  stress  t  and  the  axis. of  the  shaft,  the  angle 
between  the  plane  on  which  t  acts  and  the  axis  of  the  shaft 
will  be  90° +0. 

Under  this  method  of  treatment  it  would  be  necessary 
to  design  the  shaft  so  that  t  should  not  exceed  the  greatest 
prescribed  tensile  working  stress  for  the  material  em- 
ployed. 

The  greatest  intensity  of  oompressive  stress  in  the  shaft 
would  be  found  by  giving  the  negative  sign  to  the  radical 
in  the  second  member  of  eq.  (n). 

The  preceding  formulae  have  been  established  in  a 
manner  to  make  them  applicable  to  any  form  of  shaft 
section  or  any  values  of  k  and  T.  It  is  only  necessary  to 
insert  in  those  formulae  any  intensities  of  those  stresses 
that  may  exist.  If,  for  example,  it  were  considered  desir- 

P 

able  to  add  the  shear  — -2  due  to  the  thrust  P  to  the  tor- 

p 

sional  shear  it  would  only  be  necessary  to  take   7-f — 2 

for  T  wherever  the  latter  quantity  occurs. 

If  a  shaft  is  circular  in  section,  as  is  almost  universally 
the  case,  so  that  Ip  =?  2/,  and  if  the  shearing  effect  of  P  in 
the  section  at  B,  Fig.  i,  be  omitted,  useful  and  extremely 
simple  relations  may  be  deduced.  In  that  case  D  =  2r, 
being  the  diameter  of  the  shaft,  and  /  the  angle  ABC 


252  COMBINED  STRESS  CONDITIONS.  [Ch.  VI. 

of  Fig.  i,  M  as  before    being    the    resultant  moment,  or 
M=PXAB: 


rM  cos  /         j     -r     rM  sin  /  ,  \ 

=  -~Y~        and     T  =  — -^J-.       .     .     (13) 


By  the  substitution  of  these  values  in  eq.  (n), 


i=^(I+COS/)=^(l+COS/).         .       .        (14) 

Hence 

*\i\/r 

) ds) 


Eq.  (14)  gives,  by  the  aid  of  the  first  of  eqs.  (13), 

(16) 


The  second  of  eqs.    (13)   gives,   after  substituting  the 


1        £   r       16 
value  of  — r=-™,- 
2!     irD3 


S.iMsin/  ,     x 

V  /; 


The  substitution  of  the  values  of  T  and  k  from  eqs.  (13) 
in  eq.  (12)  gives 


Art.  47] 


COMBINED  BENDING  AND   TORSION. 


253 


2T 

tan  2|8=-r-=tan  /;    /. 


y.    .    .    .    (18) . 


This  last  set  of  results  relating  to  circular  shafts  will, 
in  all  ordinary  cases,  supply  everything  required  for  the 
operations  of  design  or  of  investigations  regarding  con- 
ditions of  stress  in  existing  shafts. 

Eqs.  (13),  first  of  (14),  (16),  and  (18)  apply  as  they 
stand  to  square  shafts. 

The  first  method  involves  simpler  considerations  than 
the  second,  not  only  analytically,  but  also  in  respect  to 


FIG.  2. 

empirical  quantities  required  to  be  used.  The  test  pieces 
from  which  the  ultimate  resistance  of  the  material  is  de- 
termined are  always  taken  parallel  to  the  axis  of  the  shaft, 
but  the  greatest  intensity  of  stress  /  found  in  the  second 
method  has  a  direction  inclined  to  that  axis  by  the  angle  /?. 
In  general,  therefore,  it  will  probably  be  found  more 
practicable  to  use  the  first  method  rather  than  the 
second. 

In  the  case  of  the  double  crank-shaft  shown  in  Fig.  2, 
it  is  only  necessary  to  treat  each  half  precisely  as  if  it  were 
the  single  crank-arm  in  Fig.  i. 


254  COMBINED  STRESS   CONDITIONS.  [Ch.  VI. 

Art.  48. — Combined  Bending  and  Direct  Stress. 

There  are  a  considerable  number  of  practical  problems 
of  combined  flexure  and  direct  stress  of  .sufficient  impor- 
tance to  merit  careful  examination,  and  among  them  is  the 
flexure  of  long  columns  treated  in  Art.  24.  In  this  place 
the  particular  cases  to  be  considered  are  those  in  which  the 
bending  is  produced  by  a  uniform  load  at  right  angles  to 
the  axis  of  the  member,  or  by  eccentricity  of  longitudinal 
loading,  the  direct  stress  (or  external  force)  being  applied 
in  a  direction  parallel  to  the  same  axis.  Lower  chord 
eye-bars  and  other  horizontal  or  inclined  chord  members 
of  pin  bridges  belong  to  this  class. 

Let  Mj  represent  the  bending  moment  in  the  member 
at  that  section  where  the  deflection  is  greatest,  produced 
by  loading  at  right  angles  to  the  member's  axis  or  by 
eccentricity  in  the  application  of  the  longitudinal  loading; 
let  M/  represent  the  greatest  deflection  resulting  from  the 
total  bending  moment  and  direct  stress ;  also,  let  P  be  the 
total  direct  stress  acting  upon  the  member  whose  length 
is  /,  while  k  represents  the  greatest  intensity  of  stress  due 
to  bending  alone  and  at  the  distance  d  of  the  most  remote 
fibre  from  the  neutral  axis  of  the  section  at  which  the 
deflection  w'  is  found.  Finally,  let  A  be  the  area  of  cross- 
section  of  the  member  which,  together  with  the  moment  of 
inertia  7,  is  supposed  to  be  constant  throughout  the  entire 

p 
length;    and  let  g=-j,  the  intensity  of  uniform  stress  in 

the  member  due  to  the  direct  stress  or  force  P. 

The  resultant  maximum  bending  moment  in  the 
member  will  then  be 

M=Mi±Pix/ (i) 


Art.  49.]  EYE-BAR  SUBJECTED    TO  BENDING.  2$5 

If  P  is  tension  it  will  tend  to  pull  the  member  straight, 
thus  producing  a  moment  opposite  to  Mr  In  the  second 
member  of  eq.  (i),  therefore,  the  negative  sign  is  to  be  used 
for  a  member  in  tension  and  the  positive  sign  for  a  member 
in  compression. 

The  greatest  resultant  intensity  of  stress,  t,  in  the 
member  will  then  take  the  value  • 


.....  « 


The  quantity  r  is  the  radius  of  gyration,  so  that 

I=Ar2. 

When  the  intensity  /  is  prescribed,  the  required  area 
of  section  A  is 


These  equations  are  perfectly  general  and  may  be 
applied  to  all  cases  of  combined  bending  and  direct  stress. 

Art.  49.—  The  Eye-bar  Subjected  to  Bending  by  Its  Own  Weight 
or  Other  Vertical  Loading. 

Let  Fig.  i  represent  a  lower  chord  eye  -bar  of  a  pin- 
connected  bridge  with  the  length  /  and  carrying  the  total 
tension  P.  The  depth  of  the  bar  is  h  and  the  thickness  6, 
so  that  the  area  of  the  normal  section  is  bh.  The  bar  acts 
as  a  beam  carrying  its  own  weight  as  a  uniform  load  over 
the  span  /.  That  load  deflects  the  bar  as  a  beam  while  the 
direct  stress  of  tension  (P)  decreases  that  deflection  by 
tending  to  pull  the  bar  straight.  The  problem  is  to  deter- 


COMBINED  STRESS   CONDITIONS. 


[Ch.  VI. 


mine  the  greatest  stress  in  the  bar  and  incidentally  its  centre 
deflection. 

There  are  several  methods  of  procedure.  The  first  and 
simplest  method  is  approximate  in  its  results,  although 
sufficiently  close  for  some  purposes.  It  consists  in  treating 


FIG.  i. 

the  bending  and  direct  stresses  as  existing  independently, 
so  that  results  are  obtained  by  simply  adding  the  bending 
to  the  direct  intensities.  This  method  will  be  treated 
first. 

The  more  exact  method  consists  in  recognizing  the  bend- 
ing moment  as  the  resultant  of  those  due  to  the  transverse 
load  acting  on  the  bar  as  a  simply  supported  beam,  and  to 
tne  direct  stress  P  acting  with  the  greatest  deflection  as 
its  lever-arm. 

Approximate  Method. 

Although  reference  will  be  made  to  Fig.  i,  the  formulae 
as  written  will  be  equally  applicable  to  compression  mem- 
bers in  which  P  would  be  the  total  force  of  compression. 

If  the  total  weight  of  the  bar  or  compression  member 
is  W,  and  if  /  is  the  moment  of  inertia  of  its  cross-section 
about  the  neutral  axis,  while  k  is  the  greatest  intensity  of 
bending  stress  at  the  distance  d  from  the  same  axis,  the 
theory  of  flexure  gives 


Wl    kl 


Wld 
SI  ' 


(I) 


Art.  49.]  EYE-BAR  SUBJECTED    TO  BENDING.  257 

If  the  area  of  cross-section  is  represented  by  A,  while 
the  radius  of  gyration  is  r,  I=Ar2.  Again,  the  quantity 
I+d  is  called  the  "section  modulus,"  and  tabulated 
values  of  it  for  rolled  sections  may  be  found  in  hand-books. 
Let  m  be  that  modulus,  then  eq.  (i)  may  take  the  form 


The  intensity  of  direct  tension  is 


(3) 


Obviously  k  will  be  tension  on  the  lower  side  of  the  bar 
or  other  member  and  compression  on  the  upper  side.  The 
greatest  intensity  of  stress  in  the  piece  will  be  the  sum  of 
q  and  k.  Eqs.  (2)  and  (3)  will,  therefore,  give  the  value 
of  that  greatest  intensity,  t,  of  stress  as  follows: 

W 

(4) 


When  the  greatest  value  of  t  is  prescribed,  the  required 
area  of  section,  'A,  can  be  at  once  written  from  eq.  (4) 


In  the  case  of    an  eye-bar  with  the  cross-section  bh, 

,     h        .    d      6 
d=-  and  -~  =T.     Hence 
2  r2     h 

and 

bh  =  *- 


258  COMBINED  STRESS   CONDITIONS.  [Ch.  VI. 

If  the  bar  carries  any  other  uniform  load  than  its  own, 
it  is  only  necessary  to  make  W  represent  the  total  uniform 
load,  including  the  weight  of  the  bar  itself. 

Finally  the  direct  force  P  may  act  with  the  eccentricity  e. 
In  this  case  the  moment  Pe  produces  uniform  bending 
throughout  the  length  of  the  bar,  and  it  is  only  needful  to 

write    ( -£-  ±  Pe )    for    —    in  the    preceding  formulas,   the 


double  sign  showing  that  Pe  may  act  either  with  or  against 
the  moment  of  the  uniform  load. 

The  formulas  of  this  article  are  not  sufficiently  exact 
for  the  usual  cases  of  engineering  practice. 

Art.  50.—  The  Approximate  Method  Ordinarily  Employed. 

The  method  commonly  employed  in  practical  work  for 
the  treatment  of  compound  bending  and  direct  stress  is 
a  much  closer  approximation  than  the  preceding  method, 
although  not  exact.  :  Its  chief  feature  is  the  introduction 
of  the  bending  moment  produced  by  the  direct  or  longi- 
tudinal force  multiplied  by  the  actual  maximum  deflection. 
In  the  same  manner  the  moment  due  to  the  eccentricity 
of  the  line  of  action  of  that  force  is  introduced  wherever 
necessary. 

Eq.  (6a)  of  Art.  27  gives  the  following  expression  for 
the  deflection  w'  due  to  pure  bending  and  in  terms  of  the 
greatest  intensity  of  bending  stress  k,  a  being  a  constant 
depending,  among  other  things,  upon  the  distribution  of 
loading  : 


d) 


If  the  deflection  as  given  in  eq.  (i)  be  placed  equal  to 
each  of  the  two  parts  of  the  deflection  given  in  eq.  (21) 


Art.  50.]     APPROXIMATE  METHOD  ORDINARILY  EMPLOYED.       ,259 

of  Art.  28,  it  will  be  found  for  a  beam  simply  supported 
at  each  end  and  loaded  uniformly,  that  a  =  fg,  and  for 
the  same  beam  loaded  by  a  single  weight  only  at  the  centre 
of  the  span,  a  =  £5.  The  cases  which  occur  in  practice 
conform  nearly  to  that  of  a  load  uniformly  distributed 
over  the  length  /.  Hence  for  such  a  beam  there  is  ordi- 
narily tak^n 


The  moment  produced  by  the  direct  force  or  stress  P 
acting  with  the  lever  arm  w'  will  have  the  opposite  sign 
to  that  of  Ml  (the  moment  due  to  transverse  loading  or 
to  eccentricity),  if  the  member  is  in  tension,  but  if  the 
member  is  in  compression  those  two  moments  will  have 
the  same  sign.  The  resultant  equation  of  moments  may, 
therefore,  be  written 


(3) 


As  stated,  the  plus  sign  is  to  be  used  for  a  compression 
member  and  the  negative  sign  for  a  tension  member. 

If  the  value  of  w',  given  by  eq.  (2),  be  substituted  in 
eq.  (3),  the  following  value  of  k  will  result: 


(4) 


In  eq.  (4)  the  plus  sign  is  to  be  used  for  tension  mem- 
bers and  the  minus  sign  for  compression  members.  This 
equation  is  general  and  adapted  to  all  forms  of  cross- 
section  under  the  conditions  virtually  'assumed.  Although 
not  explicitly  stated,  it  is  essentially  assumed  that  the  ends 


260  COMBINED  STRESS  CONDITIONS.  [Ch.  VI. 

of  the  member  remain  absolutely  fixed  in  distance  apart. 
This  is  frequently  not  the  case,  especially  in  the  lower 
chord  eye-bar  of  a  pin-connected  bridge  subjected  to  direct 
tension  and  to  bending  due  to  its  own  weight,  the  bar 
usually  being  horizontal. 

If  the  ends  of  the  beam  or  member,  uniformly  loaded, 
are  fixed,  a  =3^*  when  k  is  the  greatest  intensity  of  bending 
stress  at  the  mid-point  of  the  member,  or  ^  if  k  is  the 
intensity  of  the  bending  stress  at  the  fixed  ends.  One  of 
those  values  (usually  ^)  is  to  be  substituted  therefore 
for  -fa  in  the  formulae  which  follow  when  the  fixed-end 
condition  exists. 

The  resultant  maximum  intensity  of  stress  t  in  the 
member  will  obviously  be 

t=k  +  q, (5) 

in  which  equation  q  is  the  uniform  intensity  P  +  A. 

Eq.  (4)  will  be  immediately  applicable  to  any  particu- 
lar case  by  substituting  in  it  the  values  of  7  and  Ml  for 
that  special  case. 

If  the  case  of  the  lower  chord  eye-bar  mentioned  in  a 
preceding  paragraph  be  considered,  the  total  weight  of  the 
bar  being  W,  while  b  and  h  represent  its  thickness  and 

bh3  Wl 

depth    respectively,    /= — •    and    Af1=-g-.     These    values 

substituted  in  eq.  (5)  will  give  the  desired  value  of  the 
resultant  intensity,  as  follows: 


Eq.   (6)  gives  the  value  of  the  maximum  intensity  of 
tension  in  the  extreme  lower  fibres  of  the  eye-bar  when 


Art.  50.]      APPROXIMATE  METHOD   ORDINARILY  EMPLOYED.       261 

subjected  to  the  total  direct  tension  P  and  to  the  bending 
due  to  its  own  weight. 

The  greatest  intensity  of  bending  stress  in  the  bar  is 
evidently  the  second  term  of  the  second  member  of  eq.  (6), 
and  it  has  the  following  value  if  the  weight  of  the  bar  per 

W 
unit  of  length  is  -j-  =  g,  or  if  the  weight  of  a  cubic  unit  of 

the  metal  is  i: 


P 


It  is  frequently  important  to  observe  what  depth  of 
bar  with  a  constant  area  of  cross-section,  subjected  to  a 
prescribed  working  stress,  will  give  the  maximum  bending 
stress  due  to  its  own  weight  when  the  length  is  fixed. 
That  depth  can  readily  be  determined  by  taking  the  first 
derivative  of  k,  as  given  by  eq.  (7),  with  h  as  the  variable. 

dk 

By  performing  that  operation  and  placing  JL=O*  there 

will  at  once  result 


The  value  of  h  resulting  from  an  application  of  eq.  (8) 
gives  the  depth  of  bar  which,  with  a  given  value  of  /,  will 
under  the  conditions  of  the  case  yield  the  greatest  in- 
tensity of  bending  stress  k\  it  indicates,  therefore,  a  limit 
of  depth  to  be  avoided  as  far  as  practicable. 

Steel  is  the  usual  structural  material  for  eye-bars  for 
which  E  may  be  taken  at  29,000,000.  For  this  value  of 
E,  h  will  become,  by  eq.  (8), 

h  = 

4900 


262 


COMBINED  STRESS  CONDITIONS. 


[Ch.  VI. 


P  m 

In  this  equation  q  =  -A  is  the  intensity  of  uniform  stress 
/I 

in  the  bar,  or  the  ' '  working  stress. 

By  placing  the  value  of  h,  as  given  by  eq.  (8),  in  the 
value  of  k,  eq.  (7),  there  will  result  the  maximum  possible 
bending  stress  in  a  bar  of  given  length  /  and  given  area  of 
cross-section  A : 


\/q 


(9) 


If  E  =  29, ooo, ooo  and  ^'  =  .286  Ib.  per  cubic  inch  for 
steel,  eq.  (9)  will  take  the  value,  for  the  corresponding 
values  of  h  in  the  equation  preceding  eq.  (9), 


5io/ 


(10) 


The  following  table  shows  at  a  glance  the  greatest 
possible  fibre  stresses  in  eye-bars  of  different  lengths  and 
depths  when  the  working  tensile  stresses  in  pounds  per 
square  inch  are  those  given  in  the  extreme  left-hand  column 
of  the  table : 


Length  of  Eye -bars  in  Feet. 


Working 

Tensile 

Stresses 

1  5 

20 

25 

30 

5 

A 

o 

in 

Pounds 
per 

. 

• 

ui 

• 

in 

i/> 

Square 
Inch. 

a 

•2  co 

"o. 

0) 

ll 

a 

o 

|| 

a 

V 

jj 

QJ 

if] 

|S 

0,' 

<U 

in 
ll 

•2  co 

Q 

to 

Q 

to 

Q 

to 

Q 

to 

Q 

to 

Q 

to 

Ins. 

Lbs.  p. 
Sq.  In. 

Ins. 

Lbs.  p 
Sq.  In. 

Ins. 

Lbs.  p. 
Sq.  In 

Ins. 

Lbs.  p. 
Sq.  In. 

Ins. 

Lbs.  p. 
Sq.  In. 

Ins. 

Lbs.  p. 
Sq.  In. 

8,000 

3-3 

1030 

4.4 

137° 

5-5 

1710 

6.6 

2050 

7-7 

2400 

8.8 

2740 

10,000 

3-7 

920 

4.9 

1220 

6.1 

1530 

7-3 

1840 

8.6 

2140 

9.8 

2450 

I  2.  OOO 

4.0 

840 

5-4 

I  I  20 

6-7 

1400 

8.0 

1680 

9.4 

1960 

10.  7 

2240 

14,000 

4-3 

780 

5-8 

1030 

7.2 

1290 

8.7 

1550 

10.  I 

1810 

ii.  6 

2070 

l6,OOO 

4.6 

730 

6.2 

970 

7-7 

I  2IO 

9-3 

1450 

10.8 

1690 

124 

1940 

In  using  the  preceding  formulae  it  is  to  be  remembered 
that  the  ordinary  unit  of  length,  as  well  as  the  unit  of 


Art.  51.]  EXACT  METHOD   OF   TREATMENT.  263 

cross-section,  is  the  linear  inch,  and  that  the  weight  i  of  a 
cubic  unit  will  then  be  the  weight  of  a  cubic  inch.  This 
investigation  will  yield  results  sufficiently  accurate  for  all 
the  usual  cases  of  engineering  practice,  although  it  does 
not  provide  for  the  straightening  effect  of  the  pull  P, 
except  as  producing  a  bending  moment  opposite  to  that 
of  the  uniformly  distributed  load  W. 

Allowance  for  any  other  distributed  loading  than  the 
weight  of  the  bar  itself,  and  for  any  eccentricity  of  the  line 
of  action  of  P  that  may  exist,  are  made  precisely  as  ex- 
plained in  the  two  paragraphs  following  eq.  (7)  of  Art.  49. 

Art.  51.  —  Exact   Method   of  Treating   Combined   Bending  and 

Direct  Stress. 

In  this  method  of  finding  the  results  of  direct  stress 
combined  with  bending  it  is  necessary  to  determine  an 
expression  for  the  centre  deflection  of  the  bar,  or  com- 
pression member,  considered  as  simply  supported  at  each 
end.  As  the  line  of  action  of  the  direct  stress  P  is  sup- 
posed to  coincide  with  the  original  centre  line  or  axis  of 
the  bar,  if  g  is  the  weight  per  linear  unit  of  the  latter,  the 
bending  moment  Mt  in.  the  second  member  of  eq.  (i), 
Art.  48,  becomes 


As  .  this  case  is  one  in  which  P  is  tension  the  general 
eq.  (i)  of  Art.  48  will  take  the  following  form  by  the  aid 
of  eq.  (7)  of  Art.  14: 


W 

In  this  equation  g  =  -r-  is  the  weight  per  linear  inch, 


264  COMBINED  STRESS   CONDITIONS.  [Ch.  VI. 

or  other  unit,  of  the  bar  or  member  producing  a  bending 
moment  opposite  to  that  induced  by  the  direct  stress  P 
acting  with  the  lever-arm  w' .  The  integration  indicated  in 
eq.  (i)  may  be  completed,  but  as  it  is  not  a  simple  integra- 
tion it  will  not  be  made  here.  As  the  greatest  bending 
stress  is  found  at  the  centre  of  span  the  centre  deflection 
.only  is  needed  and  a  different  procedure  may.  be  followed. 
Let  wl  represent  the  centre  deflection  of  the  member 
considered,  a  beam  simply  supported  at  each  end  and 
carrying  its  own  weight  only,  or  any  other  total  weight  W 
uniformly  distributed.  It  is  necessary  to  use  the  expres- 
sion for  the  work  performed,  or  resilience  of  the  beam  in 
being  deflected  at  the  centre  by  the  amount  wr  Eq.  (8) 
of  Art.  44  gives  that  resilience  as 

W*l* 
Resilience  = ^7 .•-.''  ( 2 ) 


In  producing  the  centre  deflection  wl  the  centre  of  gravity 
of  the  weight  W  will  descend  through  the  distance  w0  found 
by  placing  WwQ  equal  to  the  resilience  given  in  eq.  (2). 
Hence 

M3 

•     (3) 


> 
Also,  since  by  eq.  (26)  of  Art.  28  wi  =  ~ 


w       8 

-•=-;   •••  w.-Awx  .....   (4) 

Hence  the  resilience  becomes 

Resilience  =  W^wr       .     .     .     .     .     (5) 


Art.  51.]  EXACT  METHOD   OF   TREATMENT.  265 

If  the  value  of  W  in  terms  of  wl  be  taken  from  eq.  (26) 
of  Art.  28  and  substituted  in  eq.  (5), 

Resilience  =  3°'  12  '      w^  ......     (6) 

I  2  ^l 

Hence  the  resilience  of  a  bent  beam  varies  as  the  square 
of  the  centre  deflection. 

If  the  actual  centre  deflection  of  the  bar  or  member 
considered  be  w',  the  resilience  of  the  beam  when  deflected 
to  that  extent  will  be 


Resilience  =  (—    -  -_        ....     (7) 


The  curvature  of  the  bar  or  member  being  slight,  the 
lengths  (equal  to  each  other)  of  the  neutral  surface  with 
the  deflections  w'  and  w1  will  be,  if  I'  and  /x  are  the  corre- 
sponding lengths  of  span  or  horizontal  projections  of  the 
neutral  surface, 


Hence 


The  difference  /'  —  /t  represents  the  movement  toward 
or  from  each  other  of  the  two  ends  of  the  bar  or  member 
under  the  action  of  the  direct  stress  or  force  P. 

In  the  case  of  the  eye-bar,  the  pull  of  the  force  P  re- 
moves a  part  of  the  deflection  wv  and  in  so  doing  performs 
work  in  aiding  to  lift  the  weight  W  of  the  bar,  the  remainder 
of  the  work  of  lifting  W  being  performed  by  the  elastic 
efforts  of  the  bar  to  straighten  itself  from  the  deflection 


266  COMBINED  STRESS  CONDITIONS.  [Ch.  VI. 

Wj  to  w'  ,  the  latter  portion  of  the  work  being  represented 

W2l3  I       w'2\ 
by    the    quantity    -  —  gj(  T  --  if-     Hence  the    following 

equation  of  work  may  be  written, 

p  Sw-w'          W2l3  H/»\  8 

J  --     =^  —       -T 


, 

+  -  rr  J  --  2)=^  —  WV-Ttt/i.     (10) 

' 


2  3  24o  «jy          25 

The  conditions  under  which  the  work  represented  by 
eq.  (10)  is  performed  are  such  that  either  (wi  —  w^)  or 
(!£>!  +  w')  may  be  written  in  the  second  member.  The 
resulting  numerical  value  of  w'  will  be  the  same  in  both 
cases  but  affected  by  different  signs.  As  the  equation  is 
written  the  numerical  value  of  w'  will  be  negative. 

In  eq.  (10)  there  is  taken  I'  =/x  =/,  the  length  of  panel, 
which  may  be  done  with  essential  accuracy. 

Dividing  both  sides  of  eq.  (10)  by  (wl  —  wf)  and  solving 
for  wf, 


f  25  P   w, 


> 
The  deflection  wt  =   g  ^  ,.  appearing   in    eq.   (n)  is   a 

known  quantity. 

After  w'  is  determined,  the  resultant  bending  moment 
at  the  centre  of  the  bar  will  be 


(12) 


If  the  area  of  cross-section  of  the  bar  is  A,  the  maxi- 
mum intensity  of  stress  /  in  it  will  be,  by  eq.  (2)  of  Art.  48, 


««*  S+T* d3) 


Art.  51.]  EXACT  METHOD   OF   TREATMENT.  267 

Or  if  the  maximum  value  of  /  is  specified 


If  the  section  is  rectangular,  so  that  A  =  bh  and  d  =  ~ 

2 


l~bh 

and 

6M; 


When  the  depth  of  the  bar  is  small  in  comparison  with 
the  length  /,  it  may  happen  that  the  resultant  or  final  de- 
flection w'  will  be  such  as  to  make  the  bending  moment 
M'  equal  to  zero.  Or 

Wl  Wl 

M'=^-P«/=o;    .-.™'=~p.     .     .     (17) 

When  w'  found  by  eq.  (17)  is  less  than  wr  given  by 
eq.  (n),  eq.  (17)  is  to  be  employed.  This  result  shows 
that  the  bar  will  be  subject  to  no  bending,  but  that  it  will 
hang  like  a  flexible  cable.  The  conditions  thus  developed 
are  those,  which  indicate  when  a  horizontal  or  inclined  bar 
stressed  in  tension  ceases  to  act  partially  as  a  beam  and 
becomes  purely  or  wholly  a  tie. 

These  formulae  are  perfectly  general  for  all  cases  of 
bars  or  members  in  tension,  even  for  such  small  sections 
as  wire.  Their  application  to  individual  cases  will  show 
that  excessive  intensities  will  not  exist  where  simple  ten- 
sion members  are  held  under  stress  in  a  nearly  horizontal 
position. 


268  COMBINED  STRESS  CONDITIONS  [Ch.  VI. 

Art.  52.  —  Combined  Bending  and  Direct  Stress  in  Compression 

Members. 

If  the  ordinary  approximate  method  of  Art.  50  be  em- 
ployed, eq.  (4)  of  that  article  is  immediately  applicable, 
using  the  minus  sign  in  the  denominator,  P  being  the  total 
direct  stress  of  compression  and  Mx  the  bending  moment 
due  to  the  uniform  transverse  load  and  to  eccentricity  of 
the  line  of  action  of  P,  if  there  be  any.  The  greatest  in- 
tensity of  bending  stress  as  represented  by  that  formula 
would  then  be 


k 

K 


In  this  equation,  d  is  the  distance  from  the  neutra/  axis 
of  the  section  to  the  extreme  fibre  in  which  the  intensity  k 
exists. 

If  e  be  the  eccentricity  of  the  line  of  action  of  P,  and  if 
W  be  the  weight  of  the  compression  member  whose  length 
is/, 

(2) 


When  the  moment  of  P  produces  bending  of  the  same 
sign  with  the  transverse  load  W,  the  plus  sign  is  to  be  used 
in  eq.  (2),  and  the  minus  sign  when  those  moments  are 
opposite.  If  the  line  of  action  of  P  coincides  with  the 
axis  of  the  member,  the  moment  Pe  disappears  from  eq.  (2). 
Again,  if  the  member  is  vertical,  so  that  there  is  no  trans- 
verse bending  due  to  the  load  W,  when  the  line  of  action 
of  P  has  the  eccentricity  e, 

M,=Pe  .......     (3) 


Art.  52.]         COMBINED  STRESSES  IN  COMPRESSION  MEMBERS.     269 

This  latter  case  exists  very  frequently  in  the  columns 
of  buildings. 

Eq.  (i)  is  thus  seen  to  represent  the  greatest  intensity 
of  bending  stress  with  Ml  taken  from  either  eq.  (2)  or 
eq,  (3)  for  the  cases  of  transverse  loading,  no  transverse 
loading,  eccentric  longitudinal  loading,  or  any  combina- 
tion of  those  cases. 

The  resultant  intensity  of  stress,  i.e.,  the  greatest 
intensity  of  compressive  stress  in  the  entire  compression 
member,  will  be 

P  ,    M.d 

i  -I-  /     \ 

•    • 


ioE 


As  A  is  the  area  of  cross-section,  I=Ar2,  r  being  the 
radius  of  gyration  of  the  cross-section  of  the  compression 

p 
member.     If  q=-j  >  ecl-  (4)  wn"l  take  ^e  form 

t  =  P         M,d  M,d  ,  . 

Ar2  —  — F  Ar2 ^ 

i  o  h,  i  oE 

In  the  use  of  this  equation,  the  intensity  q  must  ob- 
viously never  exceed  the  working  value  given  by  the  column 
formula  employed.  Indeed,  if  there  is  suitable  eccentricity 
q  may  be  much  less  than  that  working  long  column  value. 

In  practical  operation  the  principal  use  of  eq.  (5) 
may  be  the  determination  of  the  area  of  cross-section  A 
with  some  prescribed  value  of  t.  It  is  usually  feasible  to 
assign  general  outside  dimensions  of  the  proposed  column 
section  and  that  will  enable  a  close  approximate  value  of 
r  to  be  assigned.  If,  at  the  same  time,  an  approximate 


27°  COMBINED  STRESS  CONDITIONS.  JCh.  VI. 

value  of  q  may  also  be  taken,  the  resolution  of  the  first  and 
third  members  of  eq.  (5)  will  at  once  give 

P    P      M,  d 

-~*  +  ^*'       '     '     '     • 


If,  on  the  other  hand,  such  an  assignment  of  q  may  not 
be  made,  it  will  be  necessary  to  solve  the  first  and  second 
members  of  eq.  (5),  as  a  quadratic  equation,  for  A.  Bring- 
ing both  terms  of  the  second  member  of  eq.  (5)  over  a 
common  denominator  and  solving  the  resulting  equation 
of  the  second  degree  in  the  usual  manner,  the  following 
general  value  of  A  will  be  found  : 


t  r  f 

P__     P    M,d\2 P~P 


bx'   k      "^  t~*~   tr2  ]  ~ioEtr2° 

Frequently  there  may  be  written  d=-  and  r  =  .^h, 
Hence 

-2=|  (nearly). 

h 
If,  again,  d=-  and  r  =  .s$h, 

-2  =|  (nearly). 

The  preceding  values  of  the  radius  of  gyration  r  repre- 
sented in  terms  of  the  depth  h  of  the  compression  member 
are  closely  approximate  for  practical  design  work. 

Eqs.  (6)  and  (7)  will  give  the  desired  area  of  section  of 
the  compression  member  carrying  "both  direct  stress  and 


Art.  52.]        COMBINED  STRESSES  IN  COMPRESSION  MEMBERS.       271 

bending  produced  by  transverse  loading  under  the  assump- 
tions of  the  method  ordinarily  employed.  Those  formulae 
are  sufficiently  accurate  for  their  purposes,  but  it  may  be 
desirable  to  use  the  more  exact  formulas  given  in  the  next 
section. 

Exact  Method  for  Combined  Compression  and  Bending. 

The  exact  procedure  for  combined  compression  and 
bending  is  identical  with  that  used  in  Art.  51,  the  formulas 
determined  there  simply  being  adapted  to  a  compressive 
longitudinal  force  instead  of  a  force  of  tension.  It  is  to  be 
observed,  as  in  the  case  of  the  tension  member,  that  the 
compression  member  may  be  horizontal  or  inclined,  so  as 
to  be  subjected  to  bending  either  from  its  own  weight  or 
from  some  other  form  of  loading  in  addition  to  that  weight. 
The  member  may  also  be  subjected  to  uniform  bending 
throughout  its  length  by  the  eccentric  application  of  the 
longitudinal  force  P  concurrently  with  the  preceding  cross 
bending,  or,  as  in  the  case  of  a  vertical  column  carrying 
eccentric  loading,  by  that  force  P  alone. 

It  is  essential  to  recognize  in  this  connection  that  while 
the  columns  may  occasionally  be  in  the  pin-end  condi- 
tion, usually  their  ends  are  essentially  in  a  condition  of 
at  least  partial  fixedness,  although  the  degree  of  fixed- 
ness is  indeterminate.  It  will  conduce  to  simplicity  of 
treatment  if  the  transverse  bending,  either  from  distributed 
loading  or  by  the  eccentricity  of  application  of  the  column 
load,  be  treated  as  if  the  ends  of  columns  are  hinged.  It 
has  been  shown  in  Art.  28  that  the  centre  deflection  of  a 
beam  of  given  length  and  cross-section  with  ends  simply 
supported  and  with  the  loading  uniformly  distributed  is 
five  times  as  great  as  when  the  ends  of  the  same  beam 
are  fixed.  In  the  following  analysis,  therefore,  the  bend- 


272  COMBINED  STRESS   CONDITIONS.  [Ch.  VI. 

ing  from  both  the  sources  named  may  be  considered  as 
produced  in  a  column  with  hinged  ends  by  a  total  uni- 
formly distributed  load  W,  sufficient  in  amount  to  cause 
one  fifth  of  the  actual  bending  moment  acting  on  the  col- 
umn with  ends  fixed.  In  this  manner  the  fixed  or  con- 
strained end  condition  of  the  actual  column  is  provided 
for,  while  the  simplicity  of  the  hinged  end  computations 
is  retained.  The  bending  moment  produced  by  P,  acting 
with  the  lever-arm  of  the  greatest  deflection,  will  concur 
with  the  bending  moment  produced  by  the  own  weight 
of  the  member  or  other  vertical  uniform  loading,  instead 
of  being  opposed  to  it,  as  was  the  case  with  the  tension 
member  of  Art.  51.  The  work  performed,  therefore,  by 
P  and  the  uniform  loading  W  will  be  equal  to  the  resilience 
or  elastic  work  performed  in  the  member  in  changing  the 
deflection  from  w1  to  w' ,  it  being  remembered,  in  this  case, 
that  w'  may  be  less  than  wr  Under  these  conditions, 
then,  eq.  (10)  of  Art.  51,  expressing  the  work  done  on  the 
beam  in  changing  the  deflection  from  the  w1  to  w'  will 
become  the  following,  the  second  member  representing  the 
resilience  or  the  work  done  by  the  elastic  stresses  through- 
out its  volume: 

8  P  fu/*-w*\      8  T  W2l5  /w'2       \ 

( j—^  )+  —  W(v/-w1)= Ev(— -i).     (8) 

3   2  \        /       /      25  24oEI\wi2       ] 

Dividing  both  members  of  this  equation  by  (w'  —  w^), 
then  solving  for  w',  the  following  value  of  the  latter  will 
immediately  result: 


25  P   M/J 
in  which 


, 
(9a) 


Art.  52.]        COMBINED  STRESSES  IN  COMPRESSION  MEMBERS.     273 

Having  found  the  deflection  w/,  the  general  equation 
for  the  resultant  maximum  bending  moment,  eq.  (i)  of 
Art.  48,  will  take  the  following  form,  in  which  the  coeffi- 
cient c  is  introduced  to  provide  for  fixedness  of  ends  in 
the  manner  shown  in  Prob.  4,  at  the  end  of  this  chapter. 
If  the  ends  are  hinged,  corresponding  to  the  end  condition 
of  a  beam  simply  supported,  c  =  i,  but  if  the  ends  are  fixed, 
c  may  be  taken  as  .5  : 

(10) 

In  this  equation  care  must  be  exercised  in  using  the 
double  signs,  observing  that  both  plus  signs  are  to  be  taken 
together  as  are  both  minus  signs;  also,  that  the  eccen- 
tricity e  in  a  vertical  column  is  taken  in  a  direction  opposite 
to  the  deflection  w',  in  which  case  e  is  to  be  considered 
positive  and  the  lever-arm  of  P  is  (e  +  w').  In  the  upper 
chord  of  bridges  e  may  be  given  such  value  that 

M  -=^-PO-w')=o  (nearly).  .     .     .     (n) 

In  the  case  of  vertical  columns,  like  those  in  buildings, 

Wl 

ordinarily  the   term   -^~  disappears,  leaving  the  bending 

moment  in  the  column: 

M=P(e  +  u/)  ......     (12) 

In  the  great  majority  of  cases  w'  is  so  small  in  com- 
parison with  e  as  to  make  it  negligible,  so  that 


(13) 


These  various  values  of  the  bending  moment  M  cover 
all  that  usually  occur  in  practical  operations. 


274  COMBINED  STRESS   CONDITIONS.  [Ch.  VI. 

If,  in  accordance  with  the  preceding  notation,  t  is  the 
maximum  resultant  intensity  of  stress  in  the  member, 
there  will  result 

P     Md      i/        M 


Evidently  the  uniform  intensity  of  compressive  stress 
p 

-*  must  not  exceed  the  intensity  of  working  stress  given 
j\ 

by  a  suitable  long  column  formula.  When  the  greatest 
working  intensity  t  is  prescribed,  the  desired  area  of  cross- 
section  of  the  compression  member  will  be 


The    closely    approximate  values  of  —  2-   given  immedi- 

ately following  eq.  (7)  may  be  used  in  a  precisely  similar 
manner  in  eq.  (15),  so  as  to  simplify  the  practical  use  of 
that  equation. 

PROBLEMS  FOR  CHAPTER  VI. 

Problem  i.  —  A  steel  eye-bar  8  ins.  by  ij  ins.  in  section 
and  32  feet  long  sustains,  in  a  horizontal  position,  a  tensile 
stress  of  144,000  pounds,  i.e.,  12,000  pounds  per  square  inch. 
Find  the  greatest  bending  tensile  intensity  of  stress  and 
the  resultant  intensity  of  tensile  stress  at  its  centre  sec- 
tion by  the  ordinary  approximate  method  of  Art.  50,  and 
by  the  exact  method  of  Art.  51.  In  this  problem,  E  = 
30,000,000;  /  =  32X12  =384  ins.;  P  =  144,000  and  W  = 

_     1.5X8X8X8 
40.8X32=1306  pounds.     Also/=--  -=64. 


Art.  52.]  PROBLEMS  FOR  CHAPTER   VL  275 

By  eq.  (6)  of  Art.  50  the  resultant  intensity  of  tensi!e 
stress  required  is 

1^06X48 
£=12,000+  —- — =  12,000+  1860  =  13,860  Ibs.  per  sq.  in. 

The  centre  deflection  w{,  due  to  own  weight  only,  used 
in  the  exact  method,  is  ^  =  .5  inch.  Hence,  by  eq.  (n) 
of  Art.  51,  the  centre  defection  under  tensile  stress  is 

w1 '  =—    — r-  =  .iti  inch. 
1  +  1.67 

The  resultant  intensity  of  tensile  stress  at  the  centre 
section  of  the  eye-bar,  is  therefore, 

/  =  12,000  -\ '^ '     g-  =  i2,oco  +2208  =  14,208  Ibs.  per  sq.  in. 

1.5  X  o  X  o 

The  approximate  method,  therefore,  gives  an  intensity 
2208^—1860  =  348  pounds  per  sq.  in.  too  small. 

Problem  2. — A  horizontal  square  2  in.  by  2  in.  steel 
bar  30  ft.  long  is  subjected  to  a  tensile  stress  of  48,000 
pounds,  i.e.,  12,000  pounds  per  square  inch.  Find  the 
same  quantities  as  in  Prob.  i.  £  =  30,000,000;  £=360 
inches;  own  weight,  W  =  4o8  pounds,  and  P=  48,000 
pounds. 

By  eq.  (6)  of  Art.  35 

£  =  12,000  +  835  =  12,835  Ibs.  per  sq.  in. 
In  the  exact  method  the  centre  deflection  due  to  own 
weight  is 

wi=6.2  inches. 

Eq.  (n)  of  Art.  51  gives 

wf  =  5 . 5  5  inches. 


276  COMBINED  STRESS   CONDITIONS.  [Ch.  VI. 

On  the  other  hand,  the  criterion,  eq.  (17)  of  Art.  51, 
gives 

408X360  . 

it/  =  JT-  -5-^ —  =  •  3^25  inch. 
8X48,000 

The  bar,  therefore,  will  be  subject  to  no  bending  and 
its  stress  will  be  simply  that  of  tension,  the  centre  deflection 
being  .3825  inch.  If  the  deflection  were  sufficient  to  give 
the  bar  sensible  inclination,  it  would  be  necessary  to  mul- 
tiply the  horizontal  force  P  =  48,000  by  the  secant  of  that 
inclination  to  obtain  the  actual  tensile  stress  in  the  bar. 

The  results  given  by  the  ordinary  approximate  method 
are  thus  seen  to  be  quite  erroneous. 

Problem  3 . — A  i  .5-inch  round  steel  bar  48  ft.  long,  carry- 
ing a  tensile  stress  of  10,000  pounds  per  square  inch,  is 
inclined  at  an  angle  of  51°  to  the  horizontal.  Will  it  be 
subjected  to  any  bending,  and  what  will  be  its  centre  de- 
flection at  right  angles  to  its  axis  if  a  =  5 1°?  The  component 
of  the  bar's  weight  producing  the  deflection  named  is 
W  cos  a,  in  which  W=  288  pounds  is  the  bar's  weight; 
W  cos  a  =  224  pounds;  £  =  48  ft.  =  576  ins.  By  the  usual 
formula,  wl=  74  ins.  Eq.  (n)  of  Art.  51  then  gives  w' 
=  72  ins.;  but  eq.  (17)  of  Art.  51  gives 

,     224X576 

w '  =-z —          —  =  .oiinch. 
8X17,700 

Hence  this  latter  deflection  is  the  true  value  and  the 
bar  is  subjected  to  no  bending  in  its  stressed  condition. 

Problem  4. — A  steel  column  18  ft.  long  sustains  a  load 
of  240,000  pounds  and  carries  a  transverse  load  (i.e.,  per- 
pendicular to  its  axis)  of  300  pounds  per  linear  foot,  the 
latter  total  being  5400  pounds.  The  column  has  a  section 
like  that  shown  as  ' '  top  chord  latticed, ' '  Page  476,  Art.  81, 


Art.  52.]  PROBLEMS  FOR   CHAPTER   VI.  277 

and  it  is  composed  of  two  i5-in.  by  J-in.  web  plates,  two 
3-in.  by  3-in.  y-lb.  angles,  two  3-in.  by  4-in.  i4-lb.  angles, 
and  one  i8-in.  by  j\-m.  top  plate.  The  sectional  area  is  35 
sq.  ins.  The  moment  of  inertia  /  is  1255,  and  the  radius  of 
gyration  r  is  6.  The  loading  is  applied  to  the  latticed 
side  of  the  column,  so  that  the  eccentricity  of  application 
is  8.5  inches.  It  is  required  to  find  the  deflection  at  the 
centre  of  the  column  length,  the  bending  moment  and 
greatest  intensity  of  stress  at  the  same  section.  Also, 
if  the  area  of  section  were  not  given,  find  that  area  if  the 
greatest  allowed  intensity  of  compression  is  12,000  pounds 
per  square  inch.  The  details  of  the  column  at  top  and 
bottom  are  first  to  be  assumed  such  as  to  make  those  ends 
essentially  fixed  and  then  hinged. 

If  the  ends  of  the  column  were  hinged,  the  centre  bend- 
ing moment  would  be 

+  240,000  X  8  .  5  =  2,  185,800  in.-lbs. 


As  the  ends  of  the  column  are  first  to  be  taken  as  fixed, 
and  as  the  deflection  in  that  condition  will  be  but  one  fifth 
of  that  existing  with  ends  hinged,  it  will  be  necessary  to 
take  one  fifth  of  the  preceding  bending  moment  and  place 
it  equal  to  the  expression  for  the  centre  bending  moment 
produced  by  a  uniformly  distributed  load  acting  on  a 
column  supposed  to  be  with  hinged  ends.  If  W  represents 
that  uniformly  distributed  load, 

Wl 

-g-  =437,160  in.-lbs. 

Hence 

W  =  1  6  ,  2  oo  pounds. 

Byeq.  (pa)  of  Art.  52,  E  being  30,  000,000  and/  =  216  ins., 
wl  =  .0565  in. 


27$  COMBINED  STRESS  CONDITIONS.  [Ch.  VI. 

Remembering  that  P  =  240,000,  eq.  (9)  then  gives 

w'  =  .00093  in. 

These  deflections  are  so  small  in  comparison  with 
2  =  8.5  inches,  that  they  will  have  no  sensible  effect  upon 
the  result  and  they  may  be  neglected. 

In  consequence  gof  the  elastic  motions  of  the  members 
of  a  steel  structure  it  is  difficult  to  estimate  accurately  the 
effect  of  such  degree  of  fixedness  of  the  ends  of  a  column 
as  may  be  attained  in  an  actual  structure,  but  it  is  probable 
that  the  resulting  bending  moment  at  the  centre  of  the 
column  due  to  eccentricity  and  lateral  loading  is  not  less 
than  one  half  that  existing  with  hinged  ends,  and  that 
ratio  will  be  employed.  In  eq.  (10)  of  Art.  52,  therefore, 
c  =  .5,  and  the  bending  moment  will  be 

...     1/5400X216  \ 

M  =— (-    — ~ —  -  +  240,000 X 8.5  1  =  1,092,900  in. -Ibs. 

Hence  by  eq.   (14)  of  the  same  Article  the  greatest  inten- 
sity of  compression  will  be 

240,000     1,092,900  X  (d  =  8.5) 

35  1255 

=  6857  +  7402  =14,259  Ibs.  per  sq.  in. 

This  computation  shows  the  serious  effect  of  eccentric 
application  of  loading. 

If  the  greatest  allowed  intensity  of  compression  is 
12,000  pounds  per  square  inch,  eq.  (15)  of  Art.  52  shows 
that  the  area  of  cross  section  required  is 

1,092,900 X8.5\  • 


Art.  52.]  PROBLEMS  FOR   CHAPTER    VL  279 

The  moment  of  inertia  /  will  now  become  1481  instead 
of  1255. 

These  results  may  be  compared  with  those  of  the  ordi- 
nary approximate  method  by  finding  the  greatest  intensity 
of  compression,  t,  by  eq.  (4)  of  Art.  52,  as  follows,  after 
displacing  TV  by  ^  on  account  of  the  fixed  end  condition: 


=  12,135  Ibs.  per  sq.  in. 
There  is,  therefore,  no  material  discrepancy. 

Results  corresponding  to  the  preceding,  but  under  the 
supposition  that  the  ends  of  the  column  are  hinged,  may 
readily  be  found  as  follows: 

Wl 

-5-  =  2,185,800;     .'.  l/7  =  8i,ooo  pounds. 

o 

Hence 

w^  ==  .  2825  in. 
and 

wf  =  .0047  in. 

While  these  deflections  are  five  times  as  large  as  before, 
w'  is  still  too  small  to  affect  sensibly  the  results  and  it  will 
be  neglected.  The  bending  moment  at  the  centre  of  the 
column  will  then  be 

^400  X  2  1  6 
M  =  —  —  g—    -  +  240,000X8.5  =2,185,800  in.-lbs. 

and  the  greatest  intensity  of  compression 

240,000   2,185,800X8.5 

35        1255" 
=  6857  +  14,800  =  21,657  Ibs.  per  sq.  in. 


280  COMBINED  STRESS  CONDITIONS.  [Ch.  VI. 

If  the  greatest  allowed  intensity  of  compression  is 
12,000  pounds  per  square  inch,  the  area  of  cross-section 
becomes 

2,i85,8ooX8.5 


The  moment  of  inertia  /  will  now  become  2259  instead 
of  1255. 

Comparing  these  results  with  those  of  the  ordinary 
approximate  method  by  finding  the"  greatest  intensity  of 
compression,  t,  by  eq.  (4)  of  Art.  52, 

240,000      2,185,800X8.5 

t= — 7 \-—  -  =  12,170  Ibs.  per  sq.  in. 

63  2259-37 

This  result  is  a  close  agreement  with  the  other. 

Problem  5. — The  pin  of  a  crank-shaft  like  that  shown 
in  Fig.  i  of  Art.  47  sustains  a  maximum  thrust,  P,  of 
32,000  pounds,  the  length  of  crank,  e,  being  20  inches, 
and  the  axial  distance,  /,  between  the  centre  of  the  thrust 
and  shaft  bearings  being  18  inches.  Find  the  diameter 
of  the  steel  shaft  at  the  bearing  B  if  the  greatest  allowed 
bending  tension,  k,  is  10,000  Ibs.  per  sq.  in.  and  the  greatest 
allowed  torsional  shear,  T,  is  7000  Ibs*.  per  sq.  in. 

In  using  the  formulae  of  Art.  47  the  data  will  be  as 
follows : 

e  =  20  ins. ;  /  =  18  ins. ;  AB  =  26.9  ins. ;  tan  j  =  f  J  =  i.m 
7=48°;     cos   ^=.669;    sin   ;=.743;    ^  =  32,000  Ibs. 
&  =  10,000  Ibs.  per  sq.  in.;    T  =  7000  Ibs.  per  sq.  in. 
bending  moment  M  =  576,000  in. -Ibs. ;    twisting  mo- 
ment Mf  =640,000  in. -Ibs. 

The  first  method  of  Art.  47  gives  for  bending,  by  using 
the  first  of  eqs.  (9), 

Xi8 

=  8.34  ins. 


10,000 


PART   II.— TECHNICAL. 


CHAPTER  VII.' 
TENSION. 

Art.  53.— General  Observations. — Limit  of  Elasticity. — Yield 

Point. 

HITHERTO  certain  conditions  affecting  the  nature  of 
elastic  bodies  and  the  mode  of  applying  external  forces 
to  them,  have  been  assumed  as  the  basis  of  mathematical 
operations,  and  from  these  last  have  been  deduced  the 
formulae  to  be  adapted  to  the  use  of  the  engineer.  These 
conditions  are  never  realized  in  nature,  but  they  are 
approached  so  closely  that,  by  the  introduction  of  empiri- 
cal quantities,  the  formulas  give  results  of  sufficient  accu- 
racy for  all  engineering  purposes;  at  any  rate,  they  are 
the  only  ones  available  in  the  study  of  the  resistance  of 
materials. 

In  determining  the  quantity  called  the  "  coefficient  or 
modulus  of  elasticity,"  it  is  supposed  that  the  body  is  per- 
fectly elastic,  i.e.,  that  it  will  return  to  its  original  form  and 
volume  when  relieved  of  the  action  of  the  external  forces, 
also  that  this  "  modulus "  is  constant.  There  is  reason 
to  believe  that  no  body  known  to  the  engineer  is  either 
perfectly  elastic  or  possesses  a  perfectly  constant  modulus 
of  elasticity.  Yet  within  certain  limits,  the  deviations 
from  these  assumptions  are  not  sufficiently  great  to  vitiate 
their  great  practical  usefulness. 

281 


282  TENSION.  [Ch.  VII. 

These  limits  for  any  given  material  are  in  the  vicinity 
of  the  "  limit  of  elasticity  "  or  "  elastic  limit."  The  limit 
of  elasticity  or  elastic  limit  of  a  material  may  be  defined  as 
that  point  of  stress  below  which  the  intensity  of  stress 
divided  by  the  rate  of  strain,  i.e.,  strain  per  unit  of  length, 
is  essentially  constant.  This  point  or  limit  is  fairly  well 
defined  for  most  grades  of  structural  steel  and  for  some 
other  ductile  metals,  but  in  other  materials  like  stone  or 
timber  it  is  difficult  to  assign  any  degree  of  stress  as  the 
limit  of  elasticity.  In  such  material  the  intensity  of  stress 
divided  by  the  rate  of  strain  sometimes  fails  to  hje  constant 
at  all.  If  the  intensities  of  stress  and  rates  of  strain  for 
such  materials  be  plotted  so  as  to  exhibit  the  relation' 
between  those  quantities  the  resulting  line  will  be  found 
to  be  a  curve  without  any  point  which  can  properly  be  con- 
sidered the  limit  of  elasticity.  Frequently  when  such 
materials  are  relieved  cf  loads,  the  dimensions,  of  the 
piece  subjected  to  stress  will  not  return  to  their  original 
values. 

Between  the  extreme  limits  of  these  materials  exhibiting 
such  a  range  of  elastic  or  physical  qualities,  all  degrees  of 
imperfect  elastic  characteristics  may  be  found.  Fortu- 
nately, however,  the  structural  materials  commonly  em- 
ployed in  engineering  operations  may  be  treated  as  if 
possessing  at  least  approximately  elastic  characteristics 
sufficient  to  make  applicable  useful  formulae  based  upon 
Hooke's  Law. 

It  should  be  stated  that  some  authorities  have  given 
arbitrary  definitions  of  the  elastic  limit,  and  that  these 
definitions  have  been  much  used.  Wertheim  and  others 
have  considered  the  elastic  limit  to  be  that  force  which 
produces  a  permanent  elongation  of  0.00005  of  the  length 
of  bar.  Again,  Styffe  defines,  as  the  limit  of  elasticity, 
a  much  more  complicated  quantity.  He  considers  the 


Art.  53.]      GENERAL   OBSERVATIONS.— LIMIT  OF  ELASTICITY.      283 

external  load  to  be  gradually  increased  by  increments, 
which  may  be  constant,  and  that  each  load,  thus  attained, 
is  allowed  to  act  during  a  number  of  minutes  given  by 
taking  100  times  the  quotient  of  the  increment  divided 
by  the  load.  Then  the  "  limit  of  elasticity  "  is  "  that  load  by 
which,  when  it  has  been  operating  by  successive  small  incre- 
ments as  above  described,  there  is  produced  an  increase  in 
the  permanent  elongation  which  bears  a  ratio  to  the  length 
of  the  bar  equal  to  o.oi  (or  approximates  most  nearly  to 
o.o i )  of  the  ratio  which  the  increment  of  weight  bears  to 
the  total  load."  (Iron  and  Steel,  p.  30.) 

These  rather  artificial  expressions  for  limit  of  elasticity, 
however,  have  now  been  abandoned  in  favor  of  what  seems 
to  be  the  most  natural  value,  i.e.,  the  point  where  the  ratio 
between  intensity  of  stress  and  rate  of  strain  ceases  to  be 
essentially  constant. 

The  preceding  observations  relate  to  the  limit  of  elas- 
ticity as  determined  by  tests  of  materials  under  direct 
tension  or  compression.  Obviously,  however,  the  coeffi- 
cient or  modulus  of  elasticity  and  elastic  limit  as  well  as 
other  physical  qualities  may  be  determined  by  subjecting 
beams  to  flexure.  Observed  deflections  under  known  loads, 
which  do  not  bend  the  tested  beam  beyond  the  elastic 
limit,  will  enable  the  coefficient  of  elasticity  to  be  com- 
puted by  using  formulae  of  the  common  theory  of  flexure. 
Similarly  the  observed  increments  of  transverse  loading 
will  yield  data  from  which  the  limit  of  elasticity  may  be 
determined. 

By  precisely  similar  procedures  the  coefficient  of  elas- 
ticity and  elastic  limit  of  material  subjected  to  torsion 
may  be  found.  All  such  results  will  be  well  defined  in 
proportion  to  the  elastic  properties  of  the  materials.  If 
those  elastic  properties  are  nearly  perfect  the  results  will 
be  well  defined.  On  the  other  hand,  they  will  be  obscure 


284  TENSION.  [Ch.  VII. 

and  ill  defined  if  the  material  possesses  only  a  low  degree 
of  elastic  properties. 

Yield  Point. 

In  the  ordinary  testing  of  materials  for  engineering  pur- 
poses the  true  elastic  limit  is  not  determined.  The  true 
elastic  limit  of  any  test  piece  is  found  by  carefully  com- 
puting the  ratio  between  intensity  of  stress  and  rate  of 
strain  for  a  loading  continually  increasing  by  comparatively 
small  increments.  Such  a  procedure  is  too  slow  for  what 
may  be  termed  the  commercial  purposes  of  engineering. 
A  much  more  rapid  and  convenient  procedure  consists  in 
carefully  observing  the  scale  beam  of  the  testing  machine. 
As  the  load  is  gradually  increased  the  scale  beam  may 
easily  be  kept  in  a  horizontal  position  by  moving  the  scale 
weights  until  a  point  of  stress  in  the  specimen  is  reached 
at  which  the  beam  drops  in  consequence  of  the  relatively 
sudden  stretching  of  the  material.  This  stretching  con- 
tinues with  such  a  material  as  structural  steel  with  a  slight 
addition  of  loading,  or  none  at  all,  to  a  remarkable  extent. 
Finally,  after  much  stretching  of  the  test  piece,  the  strained 
material  appears  to  take  on  renewed  resistance,  requiring 
additional  loading  to  produce  much  elongation.  The  inten- 
sity of  stress  in  the  specimen  when  this  sudden  stretching 
begins  is  called  the  " yield  point"  or  sometimes  the  "stretch 
limit."  It  is  but  little  above  the  elastic  limit.  In  soft  or 
mild  steels,  or  in  high  structural  steel  the  yield  point  may 
not  be  more  than  two  or  three  thousand  pounds  above  the 
elastic  limit.  The  elastic  limit  itself  is  from  one-half  to 
six-tenths  the  ultimate  resistance  for  small  specimens  or 
about  one-half  the  ultimate  resistance  for  ^large  members 
like  eye-bars,  or  a  little  less  than  that  after  annealing. 

The  ease  with  which  the  yield  point  may  be  determined 
has  led  to  its  wide  use  under  the  name  of  elastic  limit  in 


Art.  54-]  ULTIMATE  RESISTANCE,  285 

much   engineering   literature,    but   the   distinction   should 
always  be  observed. 

In  the  case  of  some  structural  materials  with  erratic  or 
defective  elastic  properties,  like  some  grades  of  cast  iron, 
it  is  practically  impossible  to  find  any  well-defined  elastic 
limit  or  even  yield  point. 

Art.  54. — Ultimate  Resistance. 

After  a  piece  of  material,  subjected  to  stress,  has  passed 
its  elastic  limit,  the  strains  increase  until  failure  takes 
place.  If  the  piece  is  subjected  to  tensile  stress,  there 
will  be  some  degree  of  strain,  either  at  the  instant  of  rup- 
ture or  somewhat  before,  accompanied  by  an  intensity  of 
stress  greater  than  that  existing  in  the  piece  in  any  other 
condition.  This  greatest  intensity  of  internal  resistance 
is  called  the  "  Ultimate  Resistance." 

In  ductile  materials  this  point  of  greatest  resistance  is 
found  considerably  before  rupture;  the  strains  beyond  it 
increasing  rapidly  while  the  resistance  decreases  until 
separation  takes  place. 

These  phenomena  are  highly  marked  in  ductile  mate- 
rials like  wrought  iron  and  structural  steel,  particularly 
in  the  latter.  In  such  cases  if  the  application  of  stress  to 
the  test  piece  is  carefully  controlled  a  considerable  stretch- 
ing of  the  piece  may  be  produced  beyond  the  point  of 
ultimate  resistance  without  actually  separating  the  metal, 
the  load  per  square  inch  of  original  section  of  the  piece 
decreasing  rapidly.  It  is  not  difficult  to  obtain  such  re- 
sults with  soft  or  mild  steel. 

The  ultimate  resistances  of  different  materials '  used  in 
engineering  constructions  can  only  be  determined  by 
actual  tests,  and  they  have  been  the  objects  of  many  ex- 
periments. 


286  TENSION.  [Ch.  VII. 

It  has  been  observed  in  these  experiments  that  many 
influences  affect  the  ultimate  resistance  of  any  given 
material,  such  as  mode  of  manufacture,  condition  (an- 
nealed or  unannealed,  etc.),  size  of  normal  cross-section, 
form  of  normal  cross-section,  relative  dimensions  of  test 
piece,  shape  of  test  piece,  etc.  In  making  new  experiments 
or  drawing  deductions  from  those  already  made,  these  and 
similar  circumstances  should  all  be  carefully  considered. 


Art.  55.— Ductility. — Permanent  Set. 

One  of  the  most  important  and  valuable  characteristics 
of  any  material  is  its  "  ductility,"  or  that  property  by 
which  it  is  enabled  to  change  its  form,  beyond  the  limit 
of  elasticity,  before  failure  takes  place.  It  is  .measured 
by  the  permanent  "  set,"  or  stretch,  in  the  case  of  a  tensile 
stress,  which  the  test  piece  possesses  after  fracture ;  also, 
by  the  decrease  of  cross-section  which  the  piece  suffers  at 
the  place  of  fracture. 

In  general  terms,  i.e.,  for  any  degree  of  strain  at  which 
it  occurs,  "  permanent  set"  is  the  strain  which  remains  in 
the  piece  when  the  external  forces  cease  their  action.  It 
will  be  seen  hereafter  that  in  many  cases,  and  perhaps  all, 
permanent  set  decreases  during  a  period  of  time  imme- 
diately subsequent  to  the  removal  of  stress.  Indeed,  in 
some  cases  of  small  strains  it  is  observed  to  disappear 
entirely. 

Art.  56. — Cast  Iron. 

Modulus  of  Elasticity  and  Elastic  Limit. 

Cast  iron  is  a  metal  produced  by  fusion  without  sub- 
sequent working  such  as  forging  or  rolling.  Except  when 
made  for  special  purposes  under  conditions  of  careful  control 


Art.  56.]  CAST  IRON.  287 

of  the  elements  entering  it,  the  quality  of  the  product  is 
irregular  and  variable.  Bubbles  of  gases  not  escaping  from 
the  molten  mass  will  leave  voids  or  "  blow-holes  "  in  the 
final  product  and  carbon  exists  both  in  the  graphitic  and 
combined  condition,  but  in  varying  proportions.  The  mode 
of  production  and  the  practically  unavoidable  irregularities 
in  cooling  induce  both  variable  conditions  of  crystallization 
and  internal  stresses  which  are  sometimes  high  enough  to 
fracture  the  completed  casting. 

There  are  some  grades  of  cast  iron  like  those  formerly 
used  for  car  .wheels  and  ordnance  which  give  high  ultimate 
resistance  and  comparatively  high  moduli  of  elasticity  and 
which  exhibit  an  approximation  at  least  to  an  elastic  limit, 
although  the  latter  point  is  never  well  defined  as  in  wrought 
iron  and  steel.  The  ordinary  soft  castings  used  in  engi- 
neering practice  for  water  pipes,  machine  frames  and  other 
similar  purposes  disclose  under  test  such  erratic  properties 
that  they  cannot  be  said  to  have  either  a  well-defined 
modulus  of  elasticity  or  any  real  elastic  limit.  The  irregular 
behavior  of  cast  iron  under  stress  is  well  shown  for  different 
grades  of  the  material  by  the  stress-strain  diagrams  shown 
in  Fig.  i,  in  which  the  vertical  ordinates  are  intensities  of 
stress,  while  the  horizontal  ordinates  or  abscissas  are  the 
strains  or  elongations  per  linear  inch.  These  curves  are. 
typical  of  what  may  be  considered  good  grades  of  cast 
iron  for  their  purpose.  The  line  oq  represents  a  fair  grade 
of  ordinary  soft  cast  iron,  while  on  and  oe  belong  to  a  higher 
grade  and  od  a  still  stronger  metal  for  special  purposes. 
The  amounts  written  at  the  extreme  upper  ends  of  the 
curves  indicate  the  loads  or  stresses  per  square  inch  at 
which  the  test  specimens  failed.  The  two  curves  Of  and 
Oe  were  constructed  from  data  given  on  pages  597  and  605 
of  the  "  U.  S.  Report  of  Tests  of  Metals  and  Other  Materi- 
als "  for  1899.  These  two  cast-iron  test  specimens  were  of 


288 


TENSION. 


[Ch.  VII. 


metal  of  .superior  or  special  grades,  proposed  to  be  used 
for  ordnance  purposes,  as  is  indicated  by  the  high  ultimate 
resistances,  22,300  and  35,280  pounds  per  square  inch. 

There  is  seen  to  be  the  greatest  diversity  in  the  incli- 
nation  and    general    character  of   the   four -strain   curves 


35280  Ibs.  , 
„-'  r 


32000 


16000  Ibs. 


.001 


.003" 


.004"  PER  INCH 


FIG.  i. 


The  curve  Oe  has  a  fairly  straight  portion  ha,  the  point  a 
representing  an  intensity  of  stress  of  7000  pounds,  while 
the  point  h  represents  an  intensity  of  2000  pounds  per 
square  inch.  The  cross-sectional  area  of  this  test  speci- 
men was  i  square  inch.  The  difference  in  strains  at  the 
two  points  a  and  h,  or  for  a  range  in  intensity  of  5000  pounds, 


Art.  56].         CAST  IRON-COEFFICIENTS    OF  ELASTICITY.  289 

was    .0003    inch.     Hence   the   coefficient   of   elasticity  for 
these  data  would  be 

E-— —  -16,667,000  pounds. 
.0003 

In  the  same  manner  the  increase  of  strain  per  linear 
inch  of  test  specimen  resulting  from  increasing  the  stress 
of  2000  pounds  per  square  inch  at  k  to  8000  pounds  per 
square  inch  at  b  was  .00032.  Hence  with  these  data  the 
coefficient  of  elasticity  would  be 

_       6000 

£  =  —      _==I8,y  50,000  pounds. 
.00032 

The  strain  curve  On  is  an  extraordinary  one  for  cast  iron, 
as  it  is  straight  for  nearly  its  entire  length.  For  the  in- 
tensity of  stress  of  16,200  pounds  the  strain  or  stretch  is 
seen  to  be  .002  inch;*  hence  the  coefficient  of  elasticity 
would  be 

16,200 
E=—       -  =  8, TOO, ooo  pounds. 

.002 

The  metal  represented  by  the  strain  curve  Oq  cannot  be 
said  to  have  any  coefficient  of  elasticity  at  all,  as  no  part  of 
the  curve  is  straight.  These  instances  selected  from  a 
large  number  of  tests  are  representative  of  what  may  be 
expected  in  elastic  behavior  of  cast  iron.  As  a  rule,  the 
grades  possessing  the  higher  ultimate  resistances  exhibit 
a  more  nearly  normal  elastic  character  and  possess  what 
may  be  termed  not  very  wTell-defmed  coefficients  of  elas- 
ticity running  from  about  14,000,000  to  perhaps  18,000,000 
pounds  per  square  inch,  while  the  usual  grades  or  quanti- 
ties employed  in  engineering  castings  may  have  no  coeffi- 
cient of  elasticity  at  all  or  as  low  as  8,000,000  or  10,000,000 
pounds  per  square  inch.  In  view  of  all  experimental  data 
available  at  the  present  time  it  is  probably  about  as  neax 


290  TENSION.  [Ch.  VII. 

correct  as  practicable  to  take  the  tensile  coefficient  of  cast 
iron  for  ordinary  engineering  purposes,  as 

£  =  12,000,000     to     14,000,000  pounds, 

or  one  half  that  of  wrought  iron.  For  the" special  grades 
of  stronger  cast  iron,  such  as  are  used  for  ordnance  and 
car-wheel  purposes,  a  coefficient  or  modulus  of  16,000,000 
pounds  to  18,000,000  pounds  per  square  inch  may  be  used. 
As  is  usually  the  case  in  cast  iron,  the  elastic  limits  of 
the  curves  in  Fig.  i  are  so  ill-defined  that  they  cannot  be 
placed  with  certainty  even  on  the  curves  Of  and  Oe,  or 
scarcely  on  On,  and  not  at  all  on  curve  Oq.  If  the  points 
are  approximately  located  on  the  first  three  of  these  curves 
they  may  perhaps  be  placed  at  b  (8000  pounds  per  square 
inch),  at  a  (7000  pounds  per  square  inch),  and  at  m  (19,000 
pounds  per  square  inch) .  In  none  .of  these  cases,  however, 
can  the  metal  be  said  to  have  either  a  well-defined  limit  of 
elasticity  or  a  true  yield  point,  and  that  observation  is  in 
general  true  of  all  cast  iron. 


Resilience,   or  Work  Performed  in  Straining  Cast  Iron. 

As  the  scale  of  the  original  of  Fig.  i  was  8000  pounds 
to  each  inch  of  vertical  ordinate  and  .001  inch  to  each  inch 
or  horizontal  ordinate  or  abscissa,  and  as  the  strains  shown 
in  Fig.  i  belong  to  a  test  piece  i  inch  square  in  section  and 
i  inch  long,  each  square  inch  of  area  on  the  original  dia- 
gram between  any  one  of  the  strain  curves  and  the  axis 
of  abscissae  drawn  through  0  will  represent  SoooX.ooi  =8 
inch-pounds  of  work  performed  in  stretching  that  test 
piece.  The  strain  at  the  point  b  on  the  curve  Of  is  .00036 
inch,  as  shown  in  the  figure,  while  the  mean  intensity  of 
stress  in  producing  that  strain  is  4400  pounds.  Hence  if 


Art.  56.]  CAST  IRON.— RESILIENCE.  291 

b  represents  the  elastic  limit  the  resilience  or  work  per- 
formed in  stretching  the  metal  up  to  the  elastic  limit  of 
8000  pounds  per  square  inch  is 

4400  X  .00036  =  1.58  inch-pounds  per  cubic  inch. 

Similarly,  if  a  is  the  elastic  limit  in  the  strain  curve  Oc, 
the  total  strain  for  each  inch  in  length  of  the  test  specimen 
is  .00038  inch  and  the  mean  intensity  of  stress  is  3750 
pounds,  all  as  shown  in  Fig.  i.  Hence  the  resilience  or 
work  performed  was 

3 7 50 X. 0003 8  =  1.43  inch-pounds  per  cubic  inch. 

A  similar  computation  may  be  made  for  the  straight  por- 
tion of  the  strain  curve  On,  but  the  preceding  operations 
sufficiently  illustrate  the  procedure. 

The  total  work  performed  in  breaking  each  specimen 
may  readily  be  found  in  precisely  the  same  manner.  In 
the  case  of  the  curve  Of  the  strains  or  elongations  of  the 
specimen  were  actually  observed  only  up  to  the  point  d, 
although  failure  actually  took  place  at  /  or  at  the  intensity, 
35,280  pounds  per  square  inch.  The  part  df  of  the  curve 
is  drawn  approximately  as  a  continuation  of  the  observed 
curve  and  therefore  is  shown  as  a  broken  line.  The  area 
included  between  the  curve  Of  and  the  horizontal  ordinate 
Os,  i.e.  the  area  of  the  figure  Ofs,  is  11.97  square  inches. 
Hence  the  work  performed  in  rupturing  the  test  piece  was 

11.97  X8ooo  X.ooi  =95.76  inch-pounds  per  cubic  inch. 

Again,  in  the  case  of  the  strain  curve  Oe  the  area  of  the 
figure  Oet  is  4.69  square  inches.  The  total  work  expended, 
therefore,  in  rupturing  the  specimen  was 

4 . 69  X 8000  X  •  ooi  =  3 7. 5  inch-pounds  per  cubic  inch. 
In  the  latter  case  the  short  portion  ce  of  the  strain  curve  is 


292  TENSION.  [Ch.  VII. 

drawn  approximately,  as  the  strain  observations  ceased 
at  c.  It  is  to  be  remembered,  as  is  indicated  in  each  of 
these  cases,  that  when  the  data  apply  to  each  linear  inch 
of  test  piece  and  each  square  inch  of  sectional  area,  the 
work  computed  will  be  for  i  cubic  inch  of  material.  It 
is  only  necessary  to  multiply  by  the  number  of  cubic  inches 
in  the  test  piece  in  order  to  obtain  the  work  performed  in 
the  entire  piece. 

Ultimate  Resistance. 

The  ultimate  tensile  resistance  of  cast  iron  is  an  ex- 
ceedingly variable  quantity;  it  may  range  from  not  more 
than  8000  or  10,000  pounds  in  castings  of  indifferent  quality 
to  values  of  nearly  50,000  pounds  per  square  inch  in  such, 
special  grades  of  metal  as  those  which  have  been  used  for 
car  wheels  and  ordnance.  Cast  iron  has  passed  com- 
pletely out  of  use  for  the  manufacture  of  heavy  guns,  but 
there  are  other  ordnance  purposes  for  which  it  is  still 
used.  The  castings  usually  employed  by  civil  engineers 
are  generally  of  soft-grade  iron;  they  are  such  as  water 
pipes,  frames,  beds  of  machines,  and  other  similar  purposes 
which  do  not  require  special  grades  produced  by  special 
mixtures  of  raw  material  or  special  processes  of  manu- 
facture. The  ultimate  resistances  will,  therefore,  be  con- 
siderably less  than  those  belonging  to  ordnance  and  car- 
wheel  irons,  or  for  specially  strong  grades  of  metal.  As 
with  all  material,  the  character  of  cast  iron  affects  to  a 
great  extent  its  resistance,  i.e.,  whether  it  is  fine  or  coarse 
grained,  as  does  also  the  character  of  the  ore  from  which 
it  is  produced. 

Three  specimens  turned  down  to  a  diameter  of  about 
.625  inch  taken  from  iron  used  in  the  Boston  water  pipes 
and  broken  at  the  Warren  Foundry,  Phillipsburg,  New 


Art.  56.] 


CAST  IRON. 


293 


Jersey,   gave  the  following  ultimate  resistances  in  pounds 
per  square  inch: 


18,300, 


13,070. 


These  results  represent  fairly  the  ultimate  resistance  of 
ordinary  cast-iron  pipe  and  other  castings  commonly  used 
in  civil  engineering  practice.  It  has  sometimes  been  stated 
that  the  outer  surface  or  "  skin  "  of  iron  castings  has  a 
greater  capacity  of  resistance  to  stress  than  the  interior 
parts.  Investigations  carefully  conducted,  however,  by  the 
late  Professor  J.  B.  Johnson  and  others  do  not  show  that 
to  be  the  case.  Indeed  it  is  practically  certain  that  there 
is  no  essential  difference  between  the  resistances  of  the 
exterior  and  interior  parts  of  a  casting  unless  it  has  been 
subjected  to  some  special  treatment.  It  is  not  unlikely 
that  this  erroneous  impression  may  have  arisen  from  the 
results  of  irregular  cooling  of  castings  producing  internal 
stresses  sometimes  sufficient  to  produce  fracture. 

The  ' '  Report  of  the  Tests  of  Metals  and  Other  Mate- 
rials" at  the  United  States  Arsenal,  Watertown,  Mass., 
for  1900,  contains  a  mass  of  tensile  tests  of  pig  irons  and 
ordnance  castings  of  a  great  variety  of  grades  and  quali- 
ties, from  which  the  following  tabular  statement  of  greatest 
and  least  values  have  been  taken.  There  are  also  given 
the  results  of  two  tests  of  gear  teeth  taken  from  the  same 
source. 

TENSILE  TESTS  OF  CAST  IRON. 


Iron. 

Ultimate  Resistance.     Lbs.  per  Sq.  In. 

Greatest. 

Least. 

Pig-  • 

31,890  Fine  granular,  gray. 

33,500     " 
12,200  Fine  or  medium  granu- 
lar, gray. 

1  1,820  Coarse  granular,  dark  gray. 

14,900       "            "          "      " 
12,080   Fine    or  medium    granu- 
lar, gray. 

Ordnance 
Castings  . 
Gear  teeth. 

294  TENSION.  [Ch.  VII. 

As  a  recapitulation  there  may  be  written: 
For  ordinary  castings: 

(  12,000,000  Ibs.  per  sq.  in. 
Modulus  of  elasticity  <  to 

(  14,000,000  Ibs.  per  sq.  in. 

Ultimate  tensile  resistance,  15,000  to  18,000  Ibs.  per  sq.  in. 
For  specially  excellent  grades: 

i  16,000,000  Ibs.  per  sq.  in. 
Modulus  of  elasticity  <  .    to 

(  18,000,000  Ibs.  per  sq.  in. 
Ultimate  tensile  resistance,  20,000  to  35,000  Ibs.  per.  sq.  in. 

Tensile  working  resistances  in  pounds  per  square  inch 
may  be  taken  as  follows : 

For  water  pipes  and  other  similar  purposes : 

3000  to  3500  Ibs.  per  sq.  in. 

With  higher  grades  of  cast  iron  for  special  purposes: 
4000  to  7000  Ibs.  per  sq.  in. 

Effects  of  Remelting,  Continued  Fusion,  Repetition  of  Stress, 
and  High  Temperatures. 

The  physical  qualities  of  cast  iron  may  be  much  im- 
proved by  remelting  and  continued  fusion.  The  product 
of  the  blast  furnace  is  commercial  pig  iron.  These  pigs 
remelted,  as  in  a  cupola  furnace,  form  the  ordinary  cast- 
ings of  engineering  work.  If  this  remelting  should  be  con- 
tinued so  as  to  secure  third  or  fourth  fusion  metal  the 
resisting  properties  of  the  iron  would  be  enhanced,  but  the 
cost  would  at  the  same  time  be  materially  increased,  and 
hence  second  fusion  metal  only  is  ordinarily  used. 

Again,  experience  has  shown  that  if  molten  metal  be 
held  in  .fusion,  even  for  a  period  of  three  hours  or  more, 
its  physical  quality  continues  to  improve,  but  the  cost  of 


Art.  57.]        WROUGHT  IRON.— MODULUS  OF  ELASTICITY.  295 

such  a  procedure  renders  it  prohibitive  for  ordinary  pur- 
poses. 

Many  investigations  have  been  made  to  determine  the 
resisting  power  of  structural  materials  to  frequent  and  con- 
tinued repetition  of  stresses,  not  only  below,  but  above  the 
elastic  limit,  the  relief  from  stress  between  two  applications 
sometimes  being  partial  and  sometimes  complete.  It  has 
been  found  that  such  repeated  stresses,  when  as  high  as 
one-half  to  three-quarters  of  the  ultimate  resistance,  pro- 
duce material  fatigue  in  cast  iron  and  final  failure  much 
below  the  ordinary  ultimate  resistance  as  determined  by  a 
gradual  application  of  load.  Such  tests  have  shown  that 
cast  iron  is  somewhat  more  sensitive  to  fatigue  than  the 
ductile  structural  materials  of  higher  ultimate  resistance. 

The  effect  of  high  temperatures  upon  the  resisting 
capacity  of  cast  iron  is  not  in  general  different  from  that 
found  for  steel  and  wrought  iron.  Little,  if  any,  softening 
is  observed  until  a  temperature  of  500°  F.  is  approached, 
but  beyond  that  limit  it  is  liable  to  begin  to  lose  capacity 
of  resistance  to  a  material  extent  if  not  rapidly. 

Art.  57. — Wrought  Iron. — Modulus  of  Elasticity. — Limit  of  Elas- 
ticity and  Yield  Point.— Resilience. — Ultimate  Resistance  and 
Ductility. 

Wrought  iron  as  a  structural  material  has  been  com- 
pletely displaced  by  the*  various  grades  of  structural  steel, 
although  it  is  still  used  in  relatively  small  quantities  for 
special  purposes.  Again,  many  bridge  and  other  struc- 
tures built  of  wrought  iron  are  still  standing,  and  it  is 
essential  to  retain  a  record  of  its  physical  qualities. 

Wrought  iron  differs  fundamentally  from  steel  in  its 
manner  of  production,  as  it  is  a  product  of  the  puddling 
furnace.  A  white-hot  spongy  mass  was  brought  out  of  a 
bath  of  molten  slag  and  passed  between  rolls,  resulting  in 


296 


TENSION. 


[Ch.  VII. 


what  were  known  as  puddle  bars.  These  were  cut  in  suit- 
able lengths,  and  placed  in  rectangular  packages  or  piles 
of  proper  size  to  produce  the  finished  bar  or  beam  by  sub- 
sequent heating  and  rolling. 

This  process  of  production  gave  to  wrought  iron  a 
fibrous  internal  structure  of  much  greater  ultimate  resist- 
ance in  the  direction  of  the  fibre  than  at  right  angles  to 
the  fibre  or  direction  of  rolling,  and  this  was  true  whatever 
shape  was  produced,  such  as  plates,  beams,  bars,  etc. 

Modulus  of  Elasticity. 

The  coefficient  or  modulus  of  elasticity  of  wrought  iron 
was  determined  by  many  tests  of  both  small  and  full-size 
bars  when  it  was  the  principal  structural  material  in  bridges 
and  other  similar  structures.  The  adjoining  table  gives 
the  results  of  tests  of  four  bars  only.  The  two  i-inch  square 
bars  were  of  fine  quality  of  wrought  iron  and  were  tested 
many  years  ago  by  Eaton  Hodgkinson.  The  results  of 
tests  of  the  5 -inch  and  3 -inch  bars  are  taken  from  the 
"  Report  of  Tests  of  Metals  for  1881  "  made  on  the  large 
testing  machine  at  the  U.  S.  Arsenal,  Watertown,  Mass. 
The  table  below  gives  full  information  as  to  the  total  strain, 
gage  length  and  stress  per  square  inch  for  the  various  bars. 
If  p  is  the  stress  per  square  inch  and  /  the  strain  per  linear 
inch  of  gaged  length,  the  coefficient  of  elasticity  E  will  have 
the  value, 


Size  of  Bar, 

Gaged 
Length, 

Total  Strain, 

Stress  per 
Sq.  Inch, 

E. 

Inches. 

Inches. 

Inches. 

Pounds. 

1X1 

120 

•04556 

10,670 

28,IOI,OOO 

1X1 

120 

•043 

10,095 

28,198,000 

5.04X1.27 

80 

.029 

2O,OOO 

27,586,000 

3-05x1 

80 

.0279 

20,000 

28,674,000 

Art.  57.]          LIMIT  OF  ELASTICITY  AND   YIELD  POINT.  297 

It  will  be  observed  that  the  four  values  shown  are  more 
nearly  the  same  than  will  be  found  in  a  long  series  of  deter- 
minations in  the  early  tests  of  engineering  materials  when 
wrought  iron  was  in  general  use.  As  a  result  of  such 
determinations,  the  value  of 

E  =  26,000,000 

may  be  taken  as  a  fair  average  value  for  wrought  iron 
members  of  structures.  For  small  specimens,  or  for  some 
special  grades  of  wrought  iron,  27,000,000  or  possibly 
28,000,000  may  be  used. 

Obviously  all  values  of  E  must  be  computed  for  inten- 
sities of  stress  less  than  the  elastic  limit. 

Limit  of  Elasticity  and  Yield  Point — Resilience. 

The  limit  of  elasticity  for  wrought  iron  is  not  nearly  so 
well  denned  as  for  structural  steel.  The  diagram  Fig.  i  has 
been  constructed  from  the  test  of  the  one-inch  square 
wrought  iron  bar  with  a  gaged  length  of  10  feet  and  with  a 
load  increasing  by  small  increments.  The  horizontal  ordi- 
nates  represent  the  total  strains  in  inches,  while  the  ver- 
tical ordinates  represent  intensities  of  stress  per  square 
inch. 

That  part  of  the  curve  from  the  origin  o  to  a  is  straight 
and  its  equation  is, 

p=EL 

Above  a  the  line  begins  to  curve  and  at  e  the  curvature 
becomes  about  as  sharp  as  at  any  point.  The  point  a, 
elastic  limit,  may  be  taken  at  26,000  pounds  per  square 
inch,  while  e,  the  yield  point,  may  be  considered  as  29,000 
pounds  per  square  inch,  although  this  latter,  point  is  not 
well  denned.  Above  e  the  curve  becomes  much  less  inclined 


298 


TENSION. 


[Ch.  VII. 


to  a  horizontal  line,  showing  that  for  small  increments  of 
load  the  stretch  of  the  specimen  is  relatively  great. 

While  these  results  belong  to  one  specimen  only  of 
wrought  iron  they  are  characteristic  of  the  metal.  Approxi- 
mately the  elastic  limit  may  be  considered  half  the  ultimate 


I 

5000 

o_ 

rt-T 

' 

—— 

•= 

-£ 

= 

-^r 

f 

J 

\ 

,  — 

^H- 

40000 

^ 

'-* 

^ 

•*" 

I 

b 

^ 

-*" 

^, 

**" 

x 

3000C/ 

—  jy 

•• 

a 

1 

[26000 

] 

I 

1 

hpoo 

0 

f 

I 

L 

1 

in 

:h 

2 

in 

h( 

-S 

3 

in 

he 

s 

4 

in 

ht 

s 

g                                                                                                        h 

FIG.  i. 


resistance  and  the  yield  point  possibly  2000  to  4000  pounds 


more. 


For  all  ordinary  cases  of  wrought-iron  structures  the 
elastic  limit  may  safely  be  considered  22,000  to  24,000 
pounds  per  square  inch  and  the  yield  point  from  25,000  to 
28,000  pounds  per  square  inch,  as  it  is  to  be  remembered 
that  the  elastic  limit  and  yield  point  will  be  higher  for 
small  test  specimens  than  for  full-size  structural  mem- 
bers. 


Art.  57.]       WROUGHT  IRON.— DUCTILITY  AND  RESILIENCE.          299 

Ductility  and  Resilience. 

In  Fig.  i  the  horizontal  coordinates  of  the  stress-strain 
curve  are  the  strains  for  120  inches  in  length  of  a  wrought- 
iron  test  bar,  corresponding  at  each  point  to  the  intensities 
of  stress  per  square  inch  shown  on  the  vertical  line  through 
o.  This  curve  exhibits  fully  the  physical  characteristics  of 
the  material  under  test.  The  straight  part  oa  of  the  curve 
belongs  to  that  part  of  the  loading  below  the  elastic  limit 
a,  i.e.,  below  26,000  pounds  per  square  inch.  The  point  e 
indicates  the  stretch  limit  at  about  29,000  pounds  per 
square  inch.  There  is  no  constant  proportionality  be- 
tween stress  and  strain  above  a  nor  is  there  any  great 
increase  in  the  strain  for  a  given  small  increment  of  load- 
ing until  the  point  e  is  passed,  but  above  that  point  the 
stretch  for  each  constant  increment  of  loading  becomes 
relatively  large.  Beyond  the  point  b,  the  inclination  of 
the  stress-strain  curve  to  horizontal  is  relatively  small.  At 
or  near  c  the  curve  becomes  horizontal,  showing  the  maxi- 
mum intensity  of  resistance,  i.e.,  the  ultimate  resistance, 
and  the  broken  line  cf  indicates  a  rapidly  decreasing  load 
if  the  testing  machine  is  properly  manipulated  prior  to  the 
actual  parting  of  the  material  at  /.  Usually  the  actual 
failure  of  the  material  will  take  place  at  the  highest  point 
of  the  curve  unless  special  pains  be  taken  to  operate  the 
decrease  of  loading  and  even  under  such  conditions  the 
material  must  be  highly  ductile  to  produce  the  part  of 
the  curve  shown  by  the  broken  line. 

The  resilience  of  work  expended  below  the  elastic  limit 
a  can  readily  be  computed  by  the  aid  of  Fig.  i,  as  it  is 
represented  by  the  triangular  area  between  the  straight 
part  of  the  stress-strain  curve  and  a  vertical  line  through 
its  upper  limit.  The  strain  at  the  elastic  limit  of  26,000 
pounds  per  square  inch  is  .11744  inch.  The  average  force 


300  TENSION.  [Ch.  VII. 

acting  upon  the  specimen  up  to  the  elastic  limit  would  be 
half  the  value  of  the  latter.  Hence  the  elastic  resilience 
or  the  work  performed  on  the  specimen  up  to  the  elastic 
limit  is 

26,000 
.U744X  —      — =  1527.6  inch-pounds. 


Inasmuch  as  the  test  specimen  was  120  inches  long,  the 
elastic  resilience  of  the  bar  would  be  12.73  inch-pounds 
per  cubic  inch  of  its  volume.  Similarly,  the  area  of  the 
irregular  figure  oebch  is  4.97  square  inches,  and  as  the 
scale  of  force  is  20,000  pounds  per  linear  inch,  that  figure 
represents  4.97X20,000  pounds  =99,360  inch-pounds  of 

work;   or  -  =828  inch-pounds  of  work  per  cubic  inch 

of  volume  of  the  test  specimen.  If  this  test-bar,  therefore, 
were  to  be  broken  by  a  falling  weight  of  100  pounds,  that 
weight  would  be  required  to  fall  through  a  height  of 

09,360  ,  . 

-  =993.6  inches. 
100 

It  is  clear  from  the  figure  that  if  the  metal  possessed 
little  ductility  so  that  its  strain  curve  extended  no  further 
than  the  point  6,  the  work  ^required  to  be  expended  in 
breaking  it  would  be  very  small  compared  with  that  needed 
for  rupturing  the  actual  wrought -iron  piece.  The  effect 
of  a  falling  weight  may  represent  a  shock  or  blow,  or  be 
taken  as  the  equivalent  of  what  is  usually  called  a  suddenly 
applied  load.  These  considerations  show  why  a  ductile 
material  requiring  so  much  more  work  to  be  performed 
to  break  it  is  much  better  adapted  to  sustain  shock  than 
a  non-ductile  or  brittle  material.  The  latter  class  of 
materials  can  be  strained  so  little  before  failure  that  little 
work  is  required  to  be  expended  to  break  them. 


Art.  57-]          WROUGHT  I  RON. -ULTIMATE  RESISTANCE.  301 

Ultimate  Resistance. 

The  ultimate  resistance  of  wrought  iron  depends  to 
some  extent,  like  structural  steel,  on  the  size  of  the  test 
specimen  or  bar,  its  treatment  during  manufacture,  and 
whether  the  piece  is  tested  in  the  direction  in  which  it  was 
rolled  or  at  right  angles  to  that  direction.  Wrought  iron 
being  a  fibrous  material,  its  ultimate  resistance  is  materially 
greater  in  the  direction  of  the  fiber  than  at  right  angles  to 
that  direction  or  in  inclined  directions.  Structural  speci- 
fications usually  prescribed  that  when  used  in  tension, 
wrought  iron  should  take  its  load  parallel  to  the  direction 
of  rolling,  particularly  for  wrought-iron  plates. 

Round  and  rectangular  bars  of  wrought  iron  of  ordinary 
structural  sizes  showed  under  tests  ultimate  resistances, 
generally  varying  from  about  45,000  to  50,000  pounds  per 
square  inch,  the  smaller  values  applying  to  large  bars  and 
the  large  values  to  bars  of  small  section. 

A  series  of  tests  of  round  bars  found  in  the  "  Report  of 
the  Committee  of  the  U.  S.  Board  Appointed  to  Test  Iron, 
Steel,  and  other  Metals,  etc.,"  showed  that  the  ultimate 
resistances  ran  from  about  60,000  per  square  inch  for  f-inch 
rounds  down  to  about  46,000  to  47,000  pounds -per  square 
inch  for  bars  4  inches  in  diameter. 

The  ultimate  resistance  of  such  wrought-iron  shapes  as 
angles,  eye  bars,  channels,  tees,  and  others  were  shown  by 
many  tests  to  be  about  the  same  as  bars  and  flats  of  the 
same  quality  and  size,  i.e.,  many  test  specimens  showed 
ultimate  resistances  running  from  about  45,000  to  50,000 
pounds  per  square  inch.  If  the  shapes  or  'plates  were 
small,  so  that  the  temperature  was  relatively  low  during 
final  passes  between  the  rolls,  the  hardening  effect  of  such 
treatment  would  raise  the  ultimate  resistance  to  some 
extent,  resulting  in  higher  values  than  for  similar  shapes 


302  TENSION.  [Ch.  VII. 

of  large  section  which  suffered  less  reduction  of  temperatures 
during  the  process  of  rolling.  Thin  plates  showed  markedly 
higher  ultimate  resistances  than  thick  plates  for  this  reason. 

Ductility. 

From  what  has  been  stated  it  is  evident  that  wrought 
iron  would  show  the  greatest  final  contraction  of  fractured 
area  and  final  stretch  when  tested  in  the  direction  of  rolling 
than  in  any  other  direction.  Again  it  is  equally  clear 
that  the  percentage  of  final  stretch  would  be  materially 
greater  for  short  specimens  than  for  long  ones,  because  the 
necking-down  at  the  section  of  fracture  would  add  a  much 
greater  percentage  to  the  length  of  a  short  specimen  than 
to  a  long  one.  While  both  final  contraction  and  final 
stretch  varied  greatly  in  different  test  pieces,  it  may  be 
stated  that  for  gage  lengths  ranging  from  about  5  feet 
to  20  feet,  full-size  wrought-iron  bars  gave  a  final  con- 
traction of  20  per  cent  to  30  per  cent  and  a  final  stretch 
of  about  half  these  values. 

Test  specimens  of  plates,  angles  and  other  shapes,  the 
final  stretch  being  measured  over  a  gage  length  of  8  inches, 
would  generally  yield  about  20  per  cent  to  30  per  cent  of 
final  contraction  and  about  10  per  cent  to  20  per  cent  of 
final  stretch. 

The  preceding  may  be  considered  fairly  representative 
values  of  ductility  of  the  best  quality  of  wrought  iron  used 
in  bridge  and  other  structures.  They  show  that  the  metal 
was  highly  ductile  arid  well  adapted  to  structural  purposes, 
although  possessing  these  desirable  qualities  to  a  less  degree 
than  structural  steel. 

Fracture  of  Wrought  Iron. 

The  characteristic  fracture  of  wrought  iron  broken  in 
tension  either  directly  or  transversely  is  rather  coarsely 


Art.  58.]  STEEL— MODULUS  OF  ELASTICITY!  303 

fibrous,  not  infrequently  exhibiting  a  few  bright  granular 
spots,  which,  in  rare  cases,  may  possibly  be  crystalline. 
This  characteristic  fibrous  fracture  is  produced  by  the 
steady  application  of  load,  but  a  piece  of  wrought  iron 
will  exhibit  a  granular  fracture  if  broken  suddenly.  Many 
statements  have  been  made  that  wrought  iron  may  become 
crystalline  and  lose  both  ultimate  resistance  and  ductility 
under  certain  conditions  of  use,  but  bright  granular  fracture 
has  probably  been  mistaken  in  such  cases  for  crystalline. 

Art.  58.— Steel. 

Modulus  of  Elasticity. 

The  great  number  of  varieties  and  grades  of  steel  brings 
into  existence  a  correspondingly  great  number  of  physical 
quantities  and  coefficients  or  moduli  used  in  its  consider- 
ation in  connection  with  the  "  Resistance  of  Materials." 

Notwithstanding  the  number  of  varieties  of  steel  used 
at  the  present  time  for  engineering  purposes,  it  is  fortunate 
in  the  interests  of  simplified  computations  to  find  their 
moduli  of  elasticity  varying  so  little  that  they  may  be 
taken  as  practically  the  same.  Again,  it  is  further  fortunate 
that  the  moduli  for  tension  and  compression  also  appear 
to  be  the  same,  and  they  are  so  taken. 

That  class  of  steel  generally  to  be  considered  here 
is  included  under  the  term  "Structural  Steel,"  which 
may  be  divided  into  low,  medium,  and  high  steel.  These 
three  grades  of  structural  steel  are  mainly  based  upon  the 
amounts  of  carbon  which  they  contain.  While  each  class 
shades  insensibly  into  another  without  well-defined  limits, 
it  may  be  approximately  stated  at  least  that  low  or  soft 
steel  will  have  carbon  ranging  from  about  .1  to  .2  per  cent., 
and  that  the  carbon  in  medium  steel  will  run  from  about 
.2  to  .3  per  cent.,  while  high  steel  will  show  about  .3  to  .45 


304  '  TENSION.  [Ch.  VII. 

per  cent,  of  carbon.  The  ultimate  resistance  of  low  steel 
may  run  from  52,000  to  60,000  pounds  per  square  inch, 
medium  steel  from  60,000  to  68,000  pounds  per  square 
inch,  and  high  steel  from  68,000  to  about  76,000  pounds 
per  square  inch,  or  possibly  higher.  Experimental  inves- 
tigations have  shown  that  the  coefficient  of  elasticity  is 
essentially  the  same  for  all  grades  of  steel  used  in  construc- 
tion. This  observation  holds  true  also  for  nickel  steel, 
which  has  within  the  past  few  years  come  into  use  for 
special  structural  purposes.  -A  considerable  number  of 
tests  of  nickel-steel  specimens,  in  some  cases  containing 
3.375  per  cent,  of  nickel  with  .3  per  cent,  of  carbon  and  .73 
per  cent,  of  manganese,  given  in  the  U.  S.  Report  of  Tests 
of  Metals  for  1898  and  1899,  show  that  the  coefficient  of 
elasticity  for  this  metal  may  be  taken  at  values  ranging 
from  28,700,000  pounds  to  30,385,000  pounds  per  square 
inch.  In  other  words,  the  coefficient  of  elasticity  of  this 
nickel  steel  may  be  taken  between  the  usual  limits  for 
ordinary  structural  steel  of  28,000,000  and  30,000,000 
pounds  per  square  inch. 

Table  I  gives  a  condensed  statement  of  the  results  of 
an  extended  investigation  made  to  determine  the  "  con- 
stants "  of  structural  steel  by  Prof,  (now  President)  P.  C. 
Ricketts,  at  the  mechanical  laboratory  of  the  Rens.  Pol. 
Inst.  in  1886.  Although  these  tests  were  made  before  as 
many  varieties  and  grades  of  steel  had  been  developed  as 
at  present,  the  values  given  in  the  table  are  accurately 
characteristic  of  the  same  grades  of  structural  steel  pro- 
duced at  the  present  time,  1915.  As  no  corresponding 
determinations  have  been  made  of  such  wide  range  nor  with 
such  a  wide  scope  of  purpose  since  that  early  date,  the 
table  has  unique  value  and  is  worthy  of  careful  study. 
Although  this  table  contains  other  values  than  those  im- 
mediately desired,  the  opportunity  of  directly  comparing 


Art.  58.] 


STEEL. 
TABLE    I. 


305 


Mark. 

Per 

Cent. 
Car- 

TENSION 

Specimen. 

Pounds  per  Square  Inch. 

bon. 

Per 

Per 

Ulti- 

. 

Diam. 

Cent. 

Cent. 

Elastic 

mate 

Coefficient 

Inches. 

Reduc. 

Elong 

Limit 

Resist- 

of Elas. 

of  Area 

in  8  Ins 

ance. 

Rivet  steel  *  ... 

1  1 

•  09 

0.756 

61.7 

30.5 

39,600 

63,600 

30,039,000 

" 

£j 

0.758 

61.7 

30.5 

38,800 

63,300 

30,010,000 

•  « 

13 

'  ' 

0-757 

60.8 

28.9 

37,8oo 

63,000 

31,160,000 

"            '.'.'. 

41 

'  ' 

0-757 

65.3 

29.6 

37,800 

62,000 

31,063,000 

"            1  .  . 

'  ' 

0-758 

65-1 

29.4 

38,600 

63,200 

30,471,000 

•• 

43 

*  ( 

0-758 

62.3 

29.9 

39,400 

62,800 

29,965,000 

«« 

61 

'  • 

0.760 

61.6 

30.1 

37,4oo 

60,600 

30,456,000 

•            '  '  ' 

62 

" 

o.  760 

60.6 

29.6 

36,900 

61,300 

30,885,000 

« 

63 

1  ' 

0.760 

61  .8 

32.2 

39,100 

61  ,900 

27-335,000 

4 

81 

o.  760 

57-9 

29.2 

38,100 

62,500 

30,618,000 

« 

8-, 

'  ' 

o-  759 

62.4 

28.4 

37,ioo 

62,300 

30,172,000 

'                      '.   '.   '. 

83 

" 

0.758 

61  .0 

28.2 

36,600 

61,400 

30,424,000 

'                     '.   .   . 

101 

'  ' 

0.756 

65.7 

28..  6 

35  600 

61,700 

29  696,000 

I 

1  02 

'  ' 

0-755 

64.7 

29.0 

36,800 

61,600 

30,075,000 

*       '.  '.  '. 

103 

" 

o-754 

64.3 

29.1 

36,900        62,100 

30,371,000 

' 

31 

" 

o-757 

63-4 

27.9 

36,700    !    61,200 

30,918,000 

* 

32 

*  ( 

0.758 

64.0 

30.4 

•  37,7oo        61,900 

30,801  ,000 

*       '  .  . 

3s 

'  ' 

0-758 

64-3 

29.2 

37,100        61,800 

31,091,000 

• 

5i 

'  ' 

0-757 

51-7 

30.1 

37,8oo 

62,900 

30,032,000 

4 

52 

' 

°-  755 

49-4 

29.2 

38,500 

63,600 

31,646,000 

53 

1 

0-757 

51-2 

28.1 

37,8oo 

61,300 

30,031,000 

* 

71 

' 

o.75o 

62.x 

30.9 

36,200 

61  ,200 

30,166,000 

*       ... 

72 

1 

0-749 

60.5 

29.6 

36,800    !    62,400 

30,415,000 

73 

' 

0-751 

61.3 

31-7 

37,800    !    62,000 

30,232,000 

*       ... 

Itl 

' 

0.752 

64-3 

29-4 

36,400        62,400 

30,030,000 

.  .  . 

112 

1 

o-754 

63.0 

29.4 

36,400    i    61,700 

30,556,000 

*       ... 

H3 

' 

o-749 

62.3 

29.  2 

36,700 

62,200 

30,011,000 

.  .  . 

21 

' 

0.752 

55-i 

29.9 

37,200 

61,600 

30,210,000 

22 

| 

0-757 

53-7 

3i.o 

36,700 

60,100 

32,965,000 

Bessemer  t    •  •  • 

N? 

.  ii 

0-753 
0.748 

53-2 

60.3 

32.0 
28.4 

39,300 
41,500 

61,000 
66,600 

30,097,000 
28,950,000 

N2 

o-754 

58.3 

28.2 

41  ,400 

65,200 

29,391,000 

N3 

0.750 

57-0 

28.2 

43,400 

67,000 

29,899,000 

.  .  . 

Oi 

.  12 

o.75i 

59-7 

27-4 

4i,5oo 

65,300 

29,186,000 

O2 

o.  750 

59-2 

28.5 

41,100 

65,100 

29,252.000 

44 

O3 

|| 

0.750 

57-4 

27.0 

41,400 

65,700 

29,464,000 

.    .    . 

TI 

0.747 

57-3 

30.6 

42,000 

66,100 

29,907,000 

44                             '    '    ' 

To 

|| 

o.  750 

^50.  i 

41,900 

65.400 

29,899,000 

•  4 

?3 

0.751 

57-i 

28.7 

41,300 

65,400 

20.270,000 

I1 

•  13 

0.763 

58.1 

26.8 

48,100 

69,400 

20,706,000 

.    .   . 

S2 

o.  760 

59-5 

27.0 

47-400 

69,300 

20,500,000 

<4 

Ss 

o.  760 

56.4 

27.  i 

47,  TOO 

70,  TOO 

20,238,000 

44 

VI 

0.763 

59.1 

28.2 

42,200 

65,300 

29,430,000 

\J-2 

o.  760 

56.6 

27.6 

42,300 

65,600 

29,678,000 

4  4 

U3 

0.756 

58.3 

27.0 

42,300 

66,400 

29,390,000 

4  4 

R! 

.  16 

0.747 

54-8 

28.9 

42.000 

68,300 

30,083,000 

44 

R2 

0.745 

55-7 

27.6 

41,700 

68,500 

30,266,000 

RS 

0.745 

55-0 

27-4 

41  ,000 

68,600 

20,442,000 

4  4 

Pi 

.17 

0.746 

56.3 

27.1 

42,100' 

70,400 

20,375,000 

?2 

0.744 

57  -2 

27.4 

42,700 

70,500 

30,158,000 

Pa 

0.749 

55-8 

27.1 

41,500 

79,600 

30,784,000 

Vj 

vl6 

0.761 

40.7 

20.5 

60,900 

97,500 

29,045  ooo 

V9 

0.756 

38.5 

19.1 

60,400 

99,600 

30,236,000 

VR 

" 

0-759 

39-5 

19.4 

69,700 

99,100 

29,989,000 

T^i 

•  39 

o.  763 

39-0 

20.0 

69.500 

95,800 

30,025  ooo 

.    .    . 

Wo 

'  ' 

o.  762 

36.8 

19.2 

69,600 

96,200 

30,944,000 

W3 

0.765 

36.7 

I9.O 

69,100 

95,200 

29,291,000 

*Ooen  hearth  from  ^teclton.  Pa. 


t  From  Troy,  N.  Y. 


306 


TENSION. 
TABLE   I. — Continued. 


[Ch.  VII. 


SHEAR. 

COMPRESSION. 

Double 

Pounds  per  Square   Inch.,' 

Shear 

Ultimate 

Over 

Pounds  per  Square  Inch. 

Single  Shear. 

Double  Shear. 

Single 

Shear 

Elastic 

Coefficient  of 

Elastic 

Ultimate 

Elastic 

Ultimate 

Ultimate. 

Limit. 

Elasticity. 

Limit. 

Resist. 

Limit. 

Resist. 

39,000 

29,897,000 

39,500 

27,113,000 

39,6oo 

4  15,440 

43,6oo 

46,460 

I  .022 

30,000 

28,444,000 

41,100 

29,110,000 

41,100 

29,025,000 

34-6oo 

4S,26o 

38,200 

47,450 

1.048 

41,000 

29,045,000 

40,200 

30,045,000 

40,200 

28,853,000 

3i,5oo 

46,020 

33,8oo 

47,590 

1.034 

40,400 

29,411,000 

41,600 

30,192,000 

41  ,600 

29,302,000 

31,700 

46,910 

33,5oo 

48,390 

i  .032 

41,600 

29,216,000 

38,600 

29,013,000 

38,600 

29,963,000 

31,100 

44,780 

34,000 

46,590 

i  .040 

38,600 

29,478,000 

38,300 

29,090,000 

38,300 

29,807,000 

35,900 

44,600 

38,500 

47,350 

i  .062 

38,300 

28,961,000 

41,700 

29,630,000 

41,700 

28,941,000 

33,8oo 

46,440 

39,400 

48,890 

1-053 

41,700 

29,696,000 

30,900 

29,437,000 

40,000 

30,009,000 

33,7oo 

45,i9o 

35,700 

47,210 

1.045 

40,000 

28,730,000 

39,500 

29,005,000 

39,700 

29,740,000 

39,900 

29,963,000 

40,000 

31,433,000 

40,000 

29,782,000 

35,8oo 

46,100 

40,700 

47,2io 

i  .024 

39,700 

29,391,000 

41,800 

28,567,000 

41,700 

29,144,000 

30,500 

49,2io 

38,600 

51  ,000 

i  .036 

41,700 

28,747,000 

41,100 

28,503,000 

41,400 

29,531,000 

34,400 

51,470 

39,5oo 

5i,47o 

I  .000 

41,200 

28,730,000 

42,600 

29,162,000 

42,400 

29,210,000 

37,000 

49,740 

40,300 

50,940 

1  .024 

41  ,000 

28,635,000 

44,400 

28,070,000 

44,800 

28,729,000 

45,000 

29,025,000 

44,100 

29,281,000 

44,300 

29,830,000 

36,600 

51,000 

40,800 

5i,5io 

I  .010 

44,200 

29,324,000 

41  ,100 

28,812,000 

41,400 

29,342,000 

36,700 

51,280 

43,8oo 

52,550 

1  .025 

41  ,000 

28,666,000 

41  ,400 

28,860,000 

41,600 

29,241  ,000 

4i,5o« 

53,26o 

46,000 

53,390 

I  .002 

41,800 

29,802,000 

55,200 

29,162,000 

54,4oo 

29,454,000 

52,500 

70,190 

54,400 

29,281  ,000 

59,500 

28,602,000 

59,200 

28,981,000 

51,900 

67,760 

59,500 

29,281,000 

Art.  58.]  STEEL.  307 

different  physical  constants  from  the  same  quality  of  steel 
is  a  sufficient  reason  for  inserting  the  entire  table  at  this 
place.  All  the  test-  pieces  were  uniformly  about  three- 
quarters  of  an  inch  in  diameter,  and  the  stretch  was  in  all 
cases  measured  on  8  inches.  The  elongations  given  are  per 
cents  of  the  original  length  of  8  inches. 

The  reductions  of  area  are  the  per  cents  of  original 
sections  of  the  test  pieces  which  indicate  the  differences 
between  the  original  and  fractured  areas. 

As  indicated,  the  first  half  of  the  table  belongs  to  speci- 
mens of  open-hearth  rivet  steel  from  Steelton,  Pa.,  while 
the  second  half  contains  results  drawn  from  tests  on  a  com- 
paratively wide  range  of  metal  from  the  Bessemer  process 
of  the  Troy  Steel  and  Iron  Co.,  of  Troy,  N.  Y.  The  open- 
hearth  rivet  steel  is  all  seen  to  contain  only  .09  per  cent, 
of  carbon,  while  the  Bessemer  metal  had  carbon  varying 
from  o.i  i  per  cent,  to  0.39  per  cent.,  with  a  wide  gap 
between  0.17  and  0.36  per  cent. 

The  specimens  i1?  i2,  and  i3  were  cut  from  the  two  ends 
and  centre  of  bar  i,  and  those  subjected  to  tension  were 
located  adjacent  to  specimens  of  the  same  name  subjected 
to  compression.  Similar  observations  apply  to  other  sets 
of  specimens  affected  by  the  same  figure  or  same  letter. 
Hence  there  is  shown  in  this  table  the  relation  of  different 
physical  quantities  belonging  to  as  nearly  identically  the 
same  material  as  the  possibilities  of  the  case  admit. 

The  coefficients  of  tensile  elasticity  exhibit  unusual 
uniformity.  Those  for  the  open-hearth  steel  show  no 
variation  with  the  small  variation  in  carbon.  Although 
the  tensile  coefficients  for  the  Bessemer  steel  are  slightly 
lower  for  the  lowest  per  cents  of  carbon  than  for  the  highest, 
yet  some  of  the  lowest  coefficients  are  found  for  the  highest 
carbons,  and  it  is  difficult  to  determine  any  essential  varia- 
tion with  varying  proportions  of  that  element. 


308  TENSION.  [Ch.  VII. 

While  the  average  of  the  tensile  coefficients  is  a  very 
little  more  for  the  open  hearth  than  for  the  Bessemer  steel, 
there  is  really  no  sensible  difference  between  them.  The 
average  tensile  coefficient  may  be  taken  at  30,000,000 
pounds  per  square  inch. 

Too  much  importance  should  not  be  attached  to  the 
percentage  of  carbon  alone  in  these  specimens,  as  the 
presence  of  other  elements  not  given,  such  as  manganese, 
phosphorus,  etc.,  exert  marked  influences  on  the  physical 
characteristics  of  steel. 

The  modulus  of  elasticity  of  the  steel  wire  used  in 
the  cables  of  long  span,  stiffened  suspension  bridges  also 
has  the  value  of  about  30,000,000  pounds,  the  ultimate 
tensile  resistance  of  such  wire  varying  from  -about  200,000 
to  220,000  pounds  per  square  inch.  The  resisting  capacity 
of  this  material  is  largely  affected  by  the  process  of  cold 
drawing  in  its  manufacture,  but  the  modulus  of  elasticity 
seems  to  experience  little  or  no  effect  of  the  cold  working. 

Variation  of  Ultimate  Resistance  with  Area  of  Cross-section. 

The  ultimate  resistance  of  a  ductile  material  like  steel 
depends  to  some  extent  upon  the  area  of  cross-section  for 
a  number  of  reasons. 

Generally  the  work  put  upon  a  bar  of  small  cross-section 
in  reducing  between  the  rolls  from  the  ingot  or  bloom  to 
the  finished  bar  will  be  greater  for  a  bar  of  small  section 
than  for  a  similar  bar  of  large  section.  Other  things  being 
equal,  the  greater  amount  of  such  work  put  upon  the  mate- 
rial the  higher  will  be  its  physical  qualities,  including  the 
ultimate  resistance.  Again,  the  temperature  of  a  small  bar 
or  thin  plate  during  its  last  passes  between  the  rolls  will 
generally  be  lower  than  for  a  bar  of  larger  cross-section 
or  for  a  thicker  plate.  In  other  words,  the  slight  tendency 


Art.  58.]  STEEL.  309 

toward  cold  rolling  tends  to  enhanced  ultimate  resistance 
and  elastic  limit. 

Finally  at  the  section  of  ultimate  failure  there  is  a 
"  necking  down  "  to  the  final  reduction  of  area  of  fracture 
within  a  short  length  of  bar.  This  means  a  rather  violent 
movement  or  flow  of  molecules  of  the  material  toward  the 
axis  of  the  bar,  distinctly  greater  in  distance  for  a  larger 
bar  than  one  of  smaller  section  for  the  same  percentage  of 
final  reduction.  This  corresponds  to  a  greater  longitudinal 
separation  of  the  molecules  near  the  axis  of  the  specimen 
for  a  large  bar  than  for  a  small  one,  which  induces  a  little 
earlier  rupture  in  the  former  bar  than  in  the  latte'r. 

For  all  these  reasons  the  somewhat  smaller  ultimate 
resistance  per  square  inch  of  cross-section  is  to  be  antici- 
pated for  bars  of  large  section,  or  plates  of  greater  thickness 
than  for  bars  of  smaller  sectional  area,  or  for  thin  plates. 
This  difference,  however,  is  much  less  for  steel  bars  and 
plates  at  the  present  time  than  in  the  case  of  wrought  iron 
when  that  material  was  widely  or  even  exclusively  used  for 
structural  purposes. 

Influence  of  Shortness  of  Specimen. 

If  the  dimensions  of  a  test  specimen  are  such  as  to  make 
exceedingly  short  that  part  within  which  failure  will  occur 
if  a  test  is  carried  to  rupture,  there  is  less  opportunity  for 
the  molecules  of  the  material  to  move  in  toward  the  axis 
of  the  piece  as  failure  is  approached,  thus  preventing  an 
unrestrained  final  reduction  of  fractured  area.  The  result 
is  an  abnormal  enhancement  of  the  ultimate  resistance. 
If  the  specimen  is  exceedingly  short,  as  in  the  case  of  its 
being  made  by  a  groove,  as  shown  in  Fig.  i,  it  is  readily 
seen  that  the  planes  of  shear  indicated  lie  mostly  in  the 
enlarged  part  of  the  test  piece.  This  condition  prevents 
the  free  movement  of  the  molecules  along  the  oblique 


TENSION. 


[Ch.  VII. 


planes,  required  to  produce  the  necking  down  or  final 
reduction  of  area  of  section.  In  other  words,  the  material 
at  and  in  the  vicinity  of  the  section  of  failure  is  substan- 
tially supported  by  that  in  the  enlarged  part  of  the  piece, 
thus  enabling  the  ultimate  section  of  fracture  to  retain  an 
abnormally  large  area,  which  correspondingly  raises  the 
ultimate  resistance.  Many  tests  have  been  made  with 
wrought-iron  specimens  to  determine  the  limits  of  this 
influence  of  shortness.  These  tests  show  that  the  length 
of  the  reduced  part  of  a  test  piece  in  which  the  section  of 


FIG.  i. 


fracture  will  be  found  should  not  be  less  than  about  four 
times  the  diameter  in  any  case  and  that  with  ductile 
material  five  or  six  times  would  be  preferable.  As  a  matter 
of  actual  engineering  practice,  the  length  of  the  reduced 
part  of  a  test  piece  is  never  less  than  about  eight  to  ten 
times  the  diameter.  In  the  case  of  a  test  piece  of  rect- 
angular section,  the  length  should  not  be  less  than  five 
or  six  times  the  greatest  dimension  of  the  cross-section, 
or  preferably  six  to  eight  times  that  dimension. 

This  matter  of  influence  of  shortness  in  test  specimens 
is  of  the  utmost  importance  in  determining  the  true  ultimate . 
resistance  of  materials.     If  the  test  piece  be  too  short  the 
ultimate  resistance  will  be  unusually  high. 

Elastic  Limit,  Resilience,  and  Ultimate  Resistance. 

In  scrutinizing  the  results  of  tests  of  specimens  and 
full-size  members  of  this  section,  it  is  to  be  observed  that 


Art.  58.] 


STEEL. 


the  elastic  limit  is  almost  invariably  the  "  stretch-limit," 
or,  as  it  is  commonly  called, ' '  the  yield-point, ' '  and  not  the 
true  ' '  elastic  limit, ' '  below  which  the  ratio  between  in- 
tensity of  stress  and  rate  of  strain  is  essentially  constant. 
It  has  already  been  shown  and  stated  that  the  true  elastic 
limit  is  from  2000  to  3000  or  4000  pounds  per  square  inch 
below  the  stretch-limit  or  yield-point.  The  stretch-limit  is 
so  readily  observed  without  delaying  the  ordinary  routine 
of  testing  that  it  has  come  to  be  called,  although  erro- 
neously, the  elastic  limit,  in  spite  of  the  fact  that  it  is  a  little 
above  the  intensity  of  stress  to  which  that  term  should  be 
applied. 


0.44$C.      72770 
BRIDGE     STEEL 


FlG.    2. 

The  elastic  properties  of  three  grades  of  steel  are  ex- 
hibited graphically  in  Fig.  2.  The  curved  lines  represent 
the  tensile  strains  of  the  steel  specimens  at  the  intensities 
of  stresses  shown.  The  vertical  ordinates  are  intensities 
of  stress  and  the  horizontal  ordinates  the  rates  of  stretch, 
i.e.,  the  stretches  per  unit  of  length,  the  latter  being  drawn 
20  times  their  actual  amounts.  The  Rock  Island  Steel 


3I2 


TENSION. 


[Ch.  VII. 


belongs  to  a  specimen  of  steel  used  for  the  combined  rail- 
road and  highway  structure  across  the  Mississippi  River 
at  Rock  Island,  111.,  the  data  being  taken  from  the  U.  S. 
Report  of  Tests  of  Metals  for  1896.  The  lines  for  axle 
steel  and  nickel  steel  are  the  graphical  representations  of 
data  taken  from  the  "  U.  S.  Report  of  Tests  of  Metals' '  for 
1899.  As  in  the  previous  case,  the  horizontal  ordinates 
are  the  stretches  per  lineal  inch  shown  at  20  times  their 
actual  values.  The  figures  at  the  right-hand  extremities 
of  the  curves  are  the  ultimate  resistances  per  square  inch. 
The  elastic  limits  and  stretch- limits  or  yield-points  are 
shown  with  clear  definition.  The  remarkably  high  elastic 
limit  of  the  nickel  steel  is  well  indicated. 

By  taking  areas  first  between  the  horizontal  axis  OB 
and  the  inclined  straight  portion  of  each  line,  and  then 
between  the  same  horizontal  axis -and  the  entire  line  in 
each  case,  the  following  values  of  the  elastic  and  ultimate 
resilience  per  cubic  inch  of  each  specimen  will  be  found: 


Rock  Island 
Steel. 

Axle  -steel. 

Nickel-steel. 

Elastic  Resilience 

24  in  Ibs 

j  c     7  in    Ifog 

40  6  in  Ibs 

Ultimate       '  ' 

10  500  in  Ibs 

10  860  in  Ibs 

1  1  040  in  Ibs 

The  three  stress-strain  lines  or  curves  of  Fig.  i  illus- 
trate completely  the  physical  characteristics  of  the  various 
grades  of  steel  indicated  under  all  degrees  of  stress  Up  to 
actual  failure,  except  that  the  lines  are  carried  only  to  the 
maximum  intensities  of  stress  sustained.  If  those  lines 
were  prolonged  to  the  actual  parting  of  the  metal,  they 
would  show  rapidly  descending  portions  like  the  broken 
portion  of  the  Rock  Island  Bridge  steel  line.  That  por- 
tion of  the  curve,  however,  has  little  practical  value, 
although  considerable  scientific  interest. 

Table  I  contains  a  synopsis  of  the  valuable  series  of 


Art.  58.]  STEEL.  313 

tests  of  specimens  by  Prof.  P.  C.  Ricketts.  This  table  has 
already  been  explained  on  page  305.  The  tension  tests 
show  remarkably  uniform  results  in  elastic  limit  and  ulti- 
mate resistance,  and  characterize  a  most  excellent  mate- 
rial. With  the  exception  of  the  two  Bessemer  specimens 
containing  0.36  and  0.39  per  cent  carbon,  all  specimens 
were  of  mild  steel. 

Table  II  exhibits  results  of  tests  of  a  number  of  un- 
usually large  eye-bars  12  inches  wide  with  other  8-inch 
and  7 -inch  bars  used  in  the  Pennsylvania  Railroad  bridge 
across  the  Delaware  River  a  short,  distance  above  Phila- 
delphia and  completed  in  1896.  There  will  also  be  found 
in  the  table  tests  of  specimens  taken  from  the  same  bars, 
together  with  the  chemical  composition.  This  table  is 
interesting  as  disclosing  the  ultimate  tensile  resistance  of 
large  bars  of  mild  steel  having  the  chemical  composition 
shown.  The  decrease  in  ultimate  tensile  resistance  and 
elastic  limit  between  the  original  bar  and  the  finished  eye- 
bar,  due  to  the  process  of  manufacture  of  the  latter,  is  also 
evident  at  a  glance.  Although  the  steel  in  the  original 
bars  shows  ultimate  resistances  revealed  by  the  tests  of 
specimens  running  from  58,300  pounds  to  69,500  pounds 
per  square  inch,  no  ultimate  resistance  of  the  completed 
bars  exceeds  59,500  pounds  per  square  inch,  while  as  small 
a  value  as  52,300  pounds  per  square  inch  is  found.  This 
table  is  taken  from  the  description  of  the  Delaware  River 
bridge  by  Mr.  F.  C.  Kunz,  Assistant  to  Vice-President 
of  the  American  Bridge  Company,  Engineering  Depart- 
ment, published  at  Vienna,  1901. 

Table  III  gives  the  results  of  testing  a  remarkable 
series  of  large  steel  eye-bars.  The  table  exhibits  not 
only  the  physical  results  of  the  tests  but  the  chemical 
composition  of  the  metal  and  the  relative  results  for 
annealed  and  unannealed  bars.  The  table  was  supplied 


314 


TENSION. 


[Ch.  VIL 


TABLE   II.* 
RESULTS  OF   TESTS   OF    EYE-BARS   AND    OF  TEST   SPECIMENS 


Full-size  Eye-bars. 

g 

2 

i 
1 

Sflg 

Length  Lt 
in  Inches. 

Length  L2 
in  Inches. 

|l 

Inch. 

Cross  -sec- 

i 
g 

HI 

•8~ 

tions  xy 

$  w  £ 

in  Inches. 

o 

,3 

c 

f  -8* 

Q->       r/i       *H 

S  s  £ 

Origi- 
nal. 

Final. 

Origi- 
nal. 

Final. 

Ultimate 
Strength. 

Elastic 
Limit. 

* 

Pn 

jS  °  u 

i 

12X2$ 

9 

37 

410 

472 

360 

414 

348 

55,ooo 

29,200 

2 

12  X  2^ 

9 

407 

478 

360 

423 

434 

53,500 

25,900 

3 

I  2  X  2T| 

9 

4i 

408 

47i 

360 

416 

433 

53,750 

27,800 

4 

I  2  X  2^ 

9 

42 

422 

487 

360 

418 

280 

58,300 

30,600 

s 

I  2  X  2^5 

9 

42 

420 

485 

360 

417 

284 

57,600 

30,000 

6 

12  X  2lS 

9 

42 

407 

483 

360 

43o 

296 

56,300 

30,000 

7 

xaXif 

9 

403 

485 

348 

423 

356 

53,ooo 

29,200 

8 

I2X  if 

9 

41 

400 

470 

348 

411 

332 

58,000 

31,400 

0 

12  X  i  IB 

9 

38 

407 

471. 

360 

419 

337 

54,800 

29,800 

10 

laX-if 

9 

44 

407 

482 

360 

430 

256 

54,900 

29,300 

ii 

I  2  X  I  & 

9 

38 

400 

447 

348 

387 

414 

52,600 

29,200 

I  2 

I  2  X  1  195 

9 

415 

490 

360 

427 

266 

52,300 

29,900 

13 

I2XlT9S 

9 

42 

411 

489 

360 

430 

293 

58,500 

30,600 

it 

8Xif 

8i 

47 

428 

483 

384 

434 

446 

53,5oo 

29,900 

16 

sxiH 

7$ 

34 

328 

394 

276 

334 

351 

53,300 

29,300 

18 
19 

8X& 

8X1 

6} 

1 

46 

40 
38 

328 
328 
470 

390 
388 
545 

276 
276 
432 

332 
328 
501 

199 

324 
306 

53,ooo 
58,500 
53,300 

29,300 
34,000 
32,900 

21 

7  x  it 
7Xi* 

8} 

44 

50- 

575 

456 

525 

495 

59,500 

32,700 

*From  page  5,  "The  Delaware  River  Bridge,  Built  by  Pencoyd  Bridge  Co.,"  by  F.  C. 
Kunz,  "Allgem.  Bauzeitung,"  Heft  i,  1901. 


-La- 


by  Mr.  Henry  W.  Hodge,  C.E.,  of  the  firm  of  consulting 
engineers,  Boiler  &  Hodge,  of  New  York  City.  The  varia- 
tion in  chemical  composition  is  accounted  for  by  the  fact 
that  the  bars  tested  were  not  specimens  of  any  actual 
lot,  but  were  forged  and  broken  for  the  purpose  of  an 
investigation  to  determine  specifications  under  which 
12  and  14  inch  eye-bars  for  the  Monongahela  River  canti- 
lever bridge  should  be  manufactured.  The  machine  in 
which  the  eye -bar  heads  were  formed  was  not  of  sufficient 


Art.  58.] 


STEEL. 


315 


TABLE   II. — Continued. 

TAKEN  FROM  THE  SAME  EYE-BARS,  DELAWARE  RIVER  BRIDGE. 


Percentage  of 

Test  Specimen  8"  Long,  Approx.  i"  Square. 

| 

JJ 

05 

l| 

Chemical  Composition 
in  Per  Cent. 

Pounds  per 
Square  Inch. 

Per  Cent. 

oj 
C 

ill 

<£j 

§ 

No. 

vw 

Stretch 
Inches,  i 
Fracturec 

!! 

CflJ 

Reductio 
at  Fra 
Sec 

C. 

S. 

p. 

Mn. 

Ulti- 
mate 
St'gth. 

Elastic 
Limit. 

Str'ch. 

Re- 
duc- 
tion. 

Process  c 

48.0 

15.0 

49.1 

0.18 

0.06 

0.04 

0.49 

65,600 

33,200 

26.0 

45-4 

basic 

i 

36.0 
41  .0 

17.7 
15-7 

44-7 
38.8 





— 





;; 

3 

43-0 

16.1 

43-  2 

,, 

4 

40.0 
36.0 

15-9 
19-5 

45-3 
38.4 

0.24 

0.05 

0.03 

0.62 

68,000 

40,300 

26.75 

51-7 

•• 

6 

44.0 

21.5 

,  , 

49.0 
44.0 

18.3 
16.5 

49-3 
41.1 

0.24 

0.08 

0.06 

0.51 

69,500 

36,200 

25-25 

42.9 

acid 

9 

42.0 

19.4 

41  .  2 

o.  23 

0.04 

0.07 

0-55 

66,700 

36,300 

27-25 

55-2 

*  * 

10 

30.0 

II  .  2 

37-4 

0.21 

o  .  07 

0.06 

0.51 

63,800 

35,8oo 

27  .00 

39-5 

basic 

ii 

45-0 
47-0 

19-5 

54-  3 
5L2 

o    24 
o   17 

o  .  05 
0.07 

0.05 
0.06 

0.65 
0.58 

66,500 
65,000 

39,000 
36,800 

28.50 
26.00 

47-9 

57-3 

acid 

13 
14 

39-o 

I3-I 

44-9 

0     21 

0.04 

0.02 

0-45 

59,5oo 

33,7oo 

33.50 

45-6 

basic 

15 

40.0 

21  .  2 

41  .9 

O     2O 

0.05 

0.03 

o.  56 

60,300 

32,000 

28.00 

58.9 

ib 

33-0 

20.5 

0     20 

0.05 

0.03 

0.56 

58,300 

30,800 

29.25 

52.7 

11 

42.0 

19-7 

48.6 

1 

10.  I 

49-3 

O  .  22 

0.08 

0.06 

0-59 

62,400 

35,6oo 

30.25 

58.0 

acid 

20 

42.  c 

15-4 

45-8 

capacity  to  give  satisfactory  results,  and  hence  it  will  be 
observed  that  most  of  the  bars  broke  in  the  head  or  neck. 
The  actual  bars  for  the  structure  were  to  be  forged  in  a 
new  machine  of  greater  capacity. 

The  results  are  highly  interesting  as  indicating  what 
excellent  results  may  be  obtained  even  for  the  largest  bars 
under  satisfactory  conditions  of  manufacture. 

Shape  Steel  and  Plates. 

A  number  of  specimen  tests  were  made,  during  1899- 
1901,  of  the  open-hearth  acid  and  basic  steel  shapes  and 
plates  for  the  construction  of  the  City  Island  bridge  and 
the  1 45th  Street  bridge  across  the  Harlem  River,  both 
in  the  city  of  New  York,  under  the  direction  and  super- 


3i6 


TENSION. 


[Ch.  VII. 


TABLE    III. 

RESULTS  OF  FULL-SIZE  EYE-BAR  TESTS  ON  TRIAL  STEEL, 
MONONGAHELA  RIVER  CANTILEVER.  BOLLER  &  HODGE, 
CONSULTING  ENGINEERS. 

The  steel  was  basic  open-hearth  metal  manufactured  and  rolled  by  the 
Carnegie  Steel  Company,  1902.     All  bars  were  about  30  feet  long. 
"A"  means  annealed  and  "N"  not  annealed. 


Chemical  Composition. 

Specimen. 

Bar. 

Elas. 

Ult. 

Elonga- 

Car. 

Phos. 

Mang. 

Sulph. 

tion, 
Percent 

Reduct. 

Pounds  per  Sq.  In. 

in  8". 

12"  Xif" 

•30 

.O2I 

.62 

.026  j 

A  37380 
N  44330 

67600 
71080 

27.0 

21-5 

46.5 
40.0 

12"  Xif" 

.28 

.020 

•  54 

.022  -j 

A  32050 

N  37230 

60230 
68000 

26.5 

25.7 

54-i 
39-8 

12"  Xif" 

.26 

.03 

•  52 

.0,   | 

A  38610 

N  39700 

69700 

72400 

28.7 
27.2 

43-9 

46.  2 

12"  Xif" 

•36 

.Oig 

•  57 

.035  1 

A  31250 

N  37740 

70720 
76980 

28.5 
18.7 

45-6 
26.6 

( 

A  29140 

67120 

28.5 

45-5 

12  Xlf 

.32 

•03 

•  5Ip 

•  °3   -j 

N  35600 

72120 

25.0 

38.5 

12''  Xif" 

.28 

•03 

.46 

.04   j 

A  37050 
N  40450 

69820 
73340 

30-5 

27-5 

45-o 
44-7 

Bar. 

Full-size  Bar. 

Remarks. 

Elas. 

Ult. 

Elongation, 
Per  cent, 
in  20  Feet. 

Reduct., 
Per  cent. 

Pounds  per  Sq.  In. 

12'    Xif" 
12'    Xif" 
12'    Xif" 

12'     Xlf" 

12'    Xif" 

12'     Xlf" 

38190 
37480 
33330 
34320 
34210 
35930 

64140 
58670 
61090 
63030 
59170 
65520 

15.05 
5-7 
n-55 
7-05 
6.0 
9.56 

55-54 

Broke  n  bodv. 
'          neck, 
head. 

bodv. 
head. 



5-96 

vision  of  the  author.  Table  IV  contains  the  results  of 
a  portion  of  such  tests  for  the  quality  of  material  used. 
It  will  be  seen  that  the  specimens  were  taken  from  a 
wide  range  of  shapes  and  plates,  and  that  a  large  portion 
of  the  material  was  produced  by  the  basic  open-hearth 
process.  The  table  is  of  special  value  in  consequence 


15x2 


PHCENIX  IRON  CO. 


A  15  X  2-in.  steel  eye-bar  forged  at  the  shops  of  the  Phoenix  Bridge  Co.,  Phoenixville,  Pa. 
The  bar  developed  an  ultimate  resistance  of  50,160  Ibs.  per  sq.  in.  and  28,000  Ibs.  per 
sq.  in.  at  elastic  limit.  The  elongation  in  8  ins.  of  the  bar,  including  the  section  of 
failure,  was  25.6  per  cent,  and  the  elongation  of  the  pin-hole  was  5.2°inches.  The 
reduction  of  area  at  the  section  of  fracture  was  52.9  per  cent. 


Art.  58.] 


STEEL. 


317 


c  j 


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3*9  TENSION.  [Ch.  VII. 

of  the  wide  range  of  sections  covered  by  it,  as  well  as  for 
the  chemical  data  which  it  contains,  showing  the  percent- 
ages of  carbon,  manganese,  phosphorus,  and  sulphur 
contained  by  the  steel.  Both  the  chemical  analyses 
and  the  physical  results  indicate  that  many  of  the  shapes 
are  of  mild  steel,  while  the  remaining  portion  is  of  low 
steel. 

The  quality  of  metal  either  in  steel  shapes  or  plates 
depends  largely  upon  the  amount  of  reduction  reached 
in  the  passage  of  the  blooms  through  the  rolls  before  the 
final  area  of  section  is  attained.  In  the  early  days  of 
rolling  steel  sufficient  work  between  the  rolls  was  not 
always  done,  and  the  quality  of  the  metal  suffered  corre- 
spondingly. This  defect  is  seldom  or  never  found  at  the 
present  time  and  the  corresponding  variations  in  certain 
physical  qualities  are  avoided.  In  the  case  of  wide  and 
thin  plates,  in  which  the  temperature  of  the  metal  may 
be  lower  than  in  thicker  plates  at  the  last  pass  through 
the  rolls,  increased  hardness  may  sometimes  be  found, 
but  as  a  rule  there  will  be  little,  if  any,  difference,  as  the 
preceding  tables  show,  in  the  physical  results  for  the  thick 
and  thin  sections  ordinarily  used  in  engineering  construction. 

Tests  of  specimens  from  a  large  variety  of  shapes, 
plates,  and  bars  used  in  the  towers  and  stiffening  trusses 
of  the  Manhattan  Suspension  Bridge  across  the  East  River 
at  New  York  City,  as  given  in  the  Report  of  Mr.  Ralph 
Modjeski,  consulting  engineer,  1909,  show  the  following 
results : 

Carbon  Steel  for  Towers. 

Metal  from  plates,  bars,  channels,  bulb  angles, 
and  rivet  rounds  gave  average  elastic  limits  for 
different  sets  of  tests  varying  from  a  maximum  of 
43,040  pounds  per  square  inch  down  to  31,137 


Art.  58.]  STEEL.  319 

pounds  per  square  inch.  The  average  ultimate 
resistances  of  the  same  sets  of  tests  gave  a  maxi- 
mum of  65,880  pounds  per  square  inch  with  inter- 
mediate values  running  down  to  51,380  pounds 
per  square  inch.  The  smaller  of  each  of  these  sets 
of  results  belongs  to  the  low  steel  used  for  rivets ; 
the  higher  values  belong  to  shapes,  plates  and  bars. 

Carbon  Steel  for  Suspended  Structures. 

Tests  of  specimens  cut  from  shapes,  plates,  bars 
and  rivet  rounds  gave  average  elastic  limits  run- 
ning from  a  maximum  of  44,505  pounds  per  square 
inch  down  to  33,907  pounds  per  square  inch.  The 
corresponding  ultimate  resistances  varied  from 
68,652  pounds  per  square  inch  down  "to  52,411 
pounds  per  square  inch.  Again,  the  smaller  values 
are  found  for  the  low-carbon  rivet  steel. 

Nickel  Steel  for  Stiffening  Trusses. 

Tests  of  specimens  cut  from  nickel-steel  shapes, 
bars  and  rivet  rounds  used  in  the  suspended 
structure  gave  average  elastic  limits  for  different 
sets  of  tests  varying  from  a  maximum  of  61,355 
pounds  per  square  inch  down  to  a  minimum  of 
55,400  pounds.  The  corresponding  ultimate  resist- 
ances varied  from  a  maximum  of  90,760  pounds 
per  square  inch  down  to  77,268  pounds  per  square 
inch. 

The  preceding  results  for  the  Manhattan  Suspension 
Bridge  show  values  which  may  reasonably  be  expected  for 
such  carbon  and  nickel  steels  as  are  now  in.  use  for  the  best 
types  of  large  bridge  structures.  The  carbon  steel  for  the 
plates,  shapes  and  bars  belongs  to  the  grade  of  medium 


322 


TENSION. 


[Ch.  VII. 


the  1 45th  Street  bridge  across  the  Harlem  River  in  New 
York  City.  The  tests  were  made  in  1901.  The  left- 
hand  column  of  the  table  shows  the-  particular  (cast) 
members  of  the  turntable  from  which  the  specimens 
were  taken.  They  also  show  that,  in -'steel  castings,  a 
sensibly  higher  grade  (in  the  sense  of  containing  more 
carbon  and  manganese)  of  steel  is  used  than  in  rolled 
shapes.  As  indicated  in  the  heading  of  the  table,  the 
material  was  acid  open-hearth  steel.  The  ultimate  tensile 
resistance  runs  from  about  67,000  pounds  to  nearly  76,000 
pounds  per  square  inch.  The  elastic  limit  is  also  observed 
to  be  high,  in  consequence  of  the  rather  large  percentage 
of  manganese.  The  quality  of  metal  exhibited  by  the 
physical  results  of  the  table  is  fairly  representative  of  that 
ordinarily  used  in  steel  castings.  Obviously  the  ductility 
exhibited  is  less  than  that  found  in  connection  with  rolled 
shapes. 

TABLE  V. 
TENSILE  TESTS  OF  ACID  OPEN-HEARTH  STEEL  CASTINGS,  1901. 


Loads  in  Pounds 
per  Sq.  In. 

II 

C^ 

oc 

Chemical  Analysis. 

Character 

Specimen  from 

ta«5 

°§ 

of 

^c 

2ti 

Fracture, 

Elastic 
Limit. 

Ulti- 
mate. 

S^ 

.S  8 

Pn£ 

C. 

Mn. 

p. 

s. 

Si. 

Turntable  wheel. 

47,510 

72,300 

31.2 

39-8 

•23 

.70 

.052 

.004 

.27 

Silky  cup. 

Track  segments.  . 

49,500 

67,200 

30.4 

29. 

.27 

•65 

.045 

.018 

.26 

«         «* 

«            (f 

47,500 

67,900 

34-3 

46.5 

.27 

.60 

.0471.007 

.28 

"      ang. 

tt            <« 

47,500 

7I,IOO 

32.0 

40.8 

•30 

•65 

.040^  .004 

.28 

Irregular. 

Rack  segments   . 

49,775 

72,H5 

31.2 

39-8 

•23 

.70 

.052 

.004 

.27 

Silky  ang. 

Track  segments  . 

47,500 

68,100 

32.6 

44-5 

.27 

.60 

.047 

.008 

:28 

"        " 

«            « 

46,270 

71,920 

29.6 

38.5 

•23 

.60 

.052  .004 

.27 

"      cup. 

«            « 

48,900 

74,700 

30.4 

42.5 

•30 

•65 

.041  .009 

.26 

Irregular. 

Turntable  wheel  . 

46,130 

71,860 

28.1 

37-6 

.29 

.70 

.046  .006 

.25 

Silky  cup. 

11                            U 

48,640 

71,600 

31-4 

39-4 

.29 

.70 

.046 

.006 

.25 

<i         « 

Rack  segments.  .  . 

49,775 

7i,335 

21.9 

37-1 

.29 

•70 

.046 

.006 

•  25 

"         " 

Track  segments.  . 

47,600  76,020 

31-6 

31-7 

•30 

•65 

.037 

.004 

.27 

ang. 

Shoes    

45,200 

68.000 

29.6 

37.7 

2S 

.  so 

.04^ 

.02 

.20 

Irregular. 

« 

47,500  68,700 

31  .  2 

O  I   •  / 

40. 

•     o 
.29 

•  o^ 

65 

'     n^O 

.05 

.024 

.27 

it 

49,800  75,7oo 

29.6 

45.5 

•31 

•  ^o 

.60 

.036 

.  -^4.1.^ 
.003 

.26 

« 

Art.  58.] 


STEEL. 


323 


Rail  Steel. 

The  grade  of  steel  adapted  to  railroad  rails  is  much 
higher  in  the  hardeners  carbon  and  manganese,  and  corre- 
spondingly higher  in  physical  quantities  than  structural 
steel,  at  the  same  time  it  is  a  quite  different  metal  from 
that  adapted  to  the  finer  purposes  of  tools ;  it  is  manufac- 
tured by  the  Bessemer  process.  The  great  increase  in 
the  immediate  past  in  the  weight  and  speed  of  railroad 
locomotives  and  trains  has  subjected  rails  to  intensely 
severe  duties  which  can  be  performed- without  deteriora- 
tion of  metal  only  by  steel  of  the  highest  powers  of  en- 
durance, which  means  a  steel  of  high  ultimate  resistance, 
elastic  limit,  and  corresponding  ductility.  The  grades 
of  steel  used  for  rail  purposes  at  the  present  time  are 
well  illustrated  by  the  following  tabular  statement,  which 
shows  the  chemical  composition  of  the  rails  of  various 
weights  and  sizes  used  by  the  N.  Y.  C.  &  H.  R.  R.  R.  Co., 
the  pounds  at  the  head  of  the  columns  indicating  the  weight 

NEW  YORK  CENTRAL  &  HUDSON  RIVER  R.  R.  SPECIFICATIONS. 


6s-Lb. 

7o-Lb. 

75-Lb. 

8o-Lb. 

ioo-Lb. 

Carbon       

O    4^ 

O/17 

O/:  - 

Silicon  

to 

0.55 

O    I  ^ 

to 
0.57 

O     I  ^ 

to 

0.60 

O     T  £ 

•55 
to 
0.60 

•°5 

to 

0.70 

to 

0.20 
I    O^ 

w.  13 
to 

O.2O 

w-  10 

to 

0.20 

•  J  5 
to 

O.2O 

0.15 

to 

0.20 

Sulphur  not  to  exceed 

to 

1.25 

o  060 

1  .  U^ 

to 

1.25 

o  060 

to 

I.30 

o  060 

to 

1-30 

to 

1.40 

Phosphorus  not  to  exceed  

o  06 

o  06 

o  06 

o  06 

Rails  having  carbon  below  will  be 
rejected  

O   A.1 

O    4^ 

o  48 

Rails  having  carbon  above  will  be 
rejected  

O    ^7 

O    ^Q 

o  62 

•53 

The  numbers  represent  the  per  cents  of  the  various  elements. 


324  TENSION.  [Ch.  VII. 

of  rail  per  yard.  The  metal  of  the  lightest  or  6  5 -pound 
rail  corresponds  to  an  ultimate  resistance  of  85,000  to  90,000 
pounds  per  square  inch,  with  an  elastic  limit  of  .5  to  .7  of 
that  value.  The  highest  or  zoo-pound  rail  corresponds  to 
metal  having  an  ultimate  tensile  resistance  of  probably 
110,000  to  120,000  pounds  per  square  inch,  with  an  elas- 
tic limit  of  .6  to  .7  of  those  amounts.  In  these  chemical 
compositions  it  is  pertinent  to  observe  the  high  carbon 
and  manganese,  and  the  low  phosphorus  and  sulphur. 

After  several  years'  experience  in  the  effort  to  secure 
a  most  enduring  steel  for  a  railroad  rail  weighing  135 
pounds  per  yard,  Mr.  James  O.  Osgood,  Chief  Engineer 
of  the  Central  Railroad  of  New  Jersey,  states  in  a  paper 
published  in  the  Official  proceedings  of  the  New  York  Rail- 
road Club  for  May  21,  1915,  that  the  following  chemical 
composition  has  yielded  the  most  satisfactory  results  within 
the  experience  of  that  road,  on  which,  where  these  heavy 
rails  are  laid,  the  traffic  is  of  excessive  intensity. 

Carbon  .85  to  i.oo  per  cent  or  carbon  .8  to  .95  per  cent. 

The  rails  having  the  latter  carbon  content  also  contain 
chromium  0.2  to  0.4  per  cent  and  nickel  0.2  to  0.4  per  cent. 
It  will  be  observed  that  this  rail  section,  i.e.,  135  pounds 
per  yard,  is  the  heaviest  yet  rolled  and  used  in  the  United 
States  up  to  the  date  of  Mr.  Osgood's  paper. 

Rivet  Steel. 

The  grade  of  steel  ordinarily  used  for  rivets  is  the 
softest,  or  lowest  in  hardeners,  employed  in  engineering 
construction;  it  should  thus  be  correspondingly  low  in 
phosphorus  and  carbon.  In  Table  I  of  this  article 
there  will  be  found  the  measures  of  ductility  and  other 
physical  properties  of  a  number  of  specimens  of  rivet 
steel,  which  are  fairly  representative  of  that  metal,  except 


Art.  58.]  STEEL.  $2$ 

that  the  ultimate  resistance  is  frequently  much  lower 
than  is  shown  there.  In  much  of  the  rivet  metal  used 
at  the  present  time  the  ultimate  tensile  resistance  may 
run  from  52,000  to  60,000  pounds  per  square  inch.  In 
such  steel  the  carbon  may  run  down  to  .06  or  .08  per  cent. 
with  sulphur  between  .02  and  .03  per  cent.,  and  phos- 
phorus even  lower.  The  treatment  to  which  rivet  metal 
must  be  subjected  in  the  heading  of  rivets  makes  it 
imperative  that  the  metal  possess  qualities  of  ductility 
and  toughness  to  an  unusual  degree  and  that  the  vari- 
ations of  temperature  in  the  rivet  shall  not  reduce  its 
resisting  capacity.  In  other  words,  rivet  steel  must  pos- 
sess physical  properties  enabling  it  to  resist  torturing 
treatment  to  the  highest  practicable  degree. 

Nickel  Steel. 

The  alloy,  nickel  steel,  to  which  the  allusion  has  already 
been  made  in  connection  with  the  subject  of  the  modulus 
of  elasticity  of  steel,  possesses  marked  characteristics  of 
high  ultimate  resistance  and  elastic  limit,  the  latter  usually 
running  from  T67  to  f  of  the  former.  The  amount  of  nickel 
in  the  alloy  is  usually  about  3.25  per  cent,  while  the  carbon 
content  may  frequently  be  .25  to  .30  per  cent,  although 
higher  values  of  the  nickel  content  will  be  found  in  the  table 
following,  which  shows  the  results  of  tests  of  both  full-size 
eye-bars  and  specimens  cut  from  those  bars.  That  table* 
vshows  the  high  ultimate  resistance  and  elastic  limit  yielded 
by  this  material,  with  but  little  if  any  decrease  in  ductility. 
The  effects  of  annealing  may  be  observed  to  be  practically 
the  same  as  for  carbon  steel. 

*  The  results  in  this  table  were  courteously  given  to  the  author  by  Mr. 
HcnryW.  Hodge,  C.  E.,  of  the  firm  of  consulting  engineers,  who  designed  and 
built  the  St.  Louis  Municipal  Bridge,  at  St.  Louis,  Mo. 


326 


TENSION. 


[Ch.  VII. 


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Art.  58.] 


STEEL. 


327 


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TENSION. 


[Ch.  VII. 


The  following  tabular  statements  give  the  physical 
qualities  of  nickel  steel  adapted  to  the  various  purposes 
indicated.  They  are  taken  from  results  published  in  the 
Railroad  Gazette  for  August  8th,  1902. 

NICKEL-STEEL  FORCINGS. 


Tensile 
Strength.Lbs. 

Elastic 
Limit,  Lbs. 

Exten., 
Per  Cent. 

Cont., 
Per  Cent 

OQ.3IO 

64,170 

2S  OO 

ST.     76 

9O,  1  40 

60,090 

2S    SO 

S4  08 

93,570 

6s,4SO 

24  oo 

4Q     -27 

Front  crank-pins    

92,l8o 

64,170 

24    SO 

si  oo 

Connecting-rods  and  guides.  .  .  . 

92,040 

59,820 

26.OO 

53.01 

NICKEL-STEEL  CASTINGS. 


Crosshead    

84,540 

5^,080 

18   so 

•7T      TO 

Furnace-bearer,  bearer-guide.  .  . 
Annealed  : 

85,050 
109,500 

54,490 
51,440 

18.00 

10    SO 

26.04 
•26    71 

Nickel  steel  

100,330 

66,720 

2S   OO 

CA    c6 

Oil-tempered  : 
Carbon  steel  

129,360 

67,230 

17    SO 

-2»       C-7 

Nickel  steel  

103,890 

76,390 

2S   OO 

61    s6 

SMALL  RIFLE  BARRELS— NICKEL  STEEL. 


Tensile  Strength,  Lbs. 

Elastic  Limit,  Lbs. 

Ext.  in  2  Inches, 
Per  Cent. 

Cont.  or  Area 
Per  Cent. 

115,100 
114,080 
"4,590 
116,620 
Il6,l2O 
"4,590 

99,820 
97,780 
99,820 
96,770 
97,780 
98,800 

23 
23 
23 
22.50 

23 
24 

64.00 
64.95 
65.45 
62.05 
64.00 
62.53 

Vanadium  Steel. 

The  alloy  called  vanadium  steel  contains  when  used  for 
many  purposes  some  chromium,  which  frequently  gives  it 
the  name  Chrome  Vanadium  Steel.  This  grade  of  steel 
contains  carbon  and  manganese  about  in  the  proportion  of 
ordinary  structural  steel.  Indeed  it  may  be  considered 


Art.  58.] 


STEEL. 


329 


ordinary  structural  steel  alloyed  with  chromium  and  vana- 
dium. The  addition  of  these  latter  materials  gives  to  the 
resulting  product  great  toughness  with  high  ultimate  resist- 
ance and  an  elastic  limit  remarkably  high  in  proportion 
to  the  ultimate  resistance.  It  is  used  largely  for  such 
special  purposes  as  locomotive  parts,  both  as  castings  and 
in  the  forged  condition.  In  either  case,  however,  it  requires 
heat  treatment.  It  is  largely  used  for  locomotive  frames, 
axles,  piston  rods,  crank  pins,  tires,  as  well  as  for  many 
parts  of  automobiles. 

Many  physical  tests  of  small  specimens  have  been  made 
giving  elastic  limits  of  about  40,000  pounds  per  square  inch 
(for  castings)  up  to  about  100,000  pounds  per  square  inch, 
the  corresponding  ultimate  tensile  resistance  being  about 
70,000  pounds  per  square  inch  up  to  about  150,000  pounds 
per  square  inch.  These  variations  in  physical  qualities 
depend  upon  chemical  contents  of  the  alloy  and  upon  the 
condition  of  the  material  as  cast  or  rolled,  and  finally  upon 
the  heat  treatment  of  the  material. 

In  a  paper  on  "  Vanadium  Steel  in  Locomotive  Con- 
struction "  by  George  L.  Norris,  Engineer  of  Tests  of  the 
American  Vanadium  Co.,  published  in  the  Official  Proceed- 
ings of  the  New  York  Railroad  Club,  1915,  he  gives  the 
following  chemical  contents  as  meeting  the  requirements 
for  the  locomotive  parts  indicated. 

Chemical  Contents  of  Chrome  Vanadium  Steel. 


Castings 

Axles,  Piston  Rods 
and  Crank  Pins 

Tires 

Carbon  

.20  to  .30% 

.30  to  .40%* 

so  to  6^% 

Manganese 

CQ  to  .70 

40  to  60 

60  to  80 

Chromium  
Silicon  
Vanadium  
Phosphorus  
Sulphur 

O 
.20  to  .30 

Over  .16 
Not  over  .05 
Not  over  05 

•75  to  1.25 
Not  over  .20 
Over  .16 
Not  over  .04 
Not  over  04 

.80  to  1.  10 

.20  tO  .35 

Over  .16 
Not  over  .05 

*  Preferred  .35% 


33° 


TENSION. 


[Ch.  VII. 


The  elastic  limit,  ultimate  resistance,  final  stretch  and 
final  reduction  of  area  corresponding  to  the  grades  of  mate- 
rial indicated  by  the  chemical  contents  are  shown  in  the 
next  table. 

PHYSICAL  REQUIREMENTS  (After  Heat  Treatment) . 


Elastic  Limit 
Lbs.   per  sq.   in. 

Ult.  Resist. 
Lbs.   per  sq.  in. 

Stretch  in 
2  ins. 

Reduction 
of  Area. 

Castings 

4O,OOO-  5O,OOO 

7O,OOO-  85,000 

25% 

45% 

Axles,       Piston 
Rods    and 
Crank  Pins.  .  . 
Tires  56"  diam. 
and  under  .... 
Tires   over   56'' 
diam  

8O,OOO-IOO,OOO 
110,000-125,000 
95,000-II5,OOO 

95,OOO-I25,OOO 
I4O,OOO-l6o,OOO 
120,000-140,000 

25 
Min.  12 
Min.  15 

55 
Min.  30 
Min.  35 

These  physical  requirements  correspond  closely  to  the 
usual  results  of  tests.  They  show  the  high  elastic  limit  of 
the  material  and  its  high  degree  of  ductility. 

Castings  must  be  carefully  annealed  by  heating  slowly 
to  about  1550°  F.  and  then  slowly  cooling. 

The  heat  treatment  for  chrome  vanadium  driving  axles 
consists  of : 

"  (i)  annealing  the  rough  forging  by  heating  carefully 
and  cooling  slowly,  (2)  reheat  ing,  "forging,  and  quenching  in 
water  or  oil,  preferably  the  latter,  (3)  then  promptly  re- 
heating slowly  and  uniformly  to  a  temperature  sufficiently 
high  to  give  the  desired  properties.  The  forging  must  be 
held  at  this  final  or  draw-back  temperature  for  at  least  two 
hours.  The  axle  should  then  be  allowed  to  cool  slowly. 

"  The  recommended  temperature  for  annealing  is  1475- 
1525°  F.,  and  for  quenching  from  1600°  F.  to  1650°  F. 
The  final  heating  for  obtaining  the  physical  properties 
should  be  approximately  1100°  F.  to  1200°  F." 

The  heat  treatment  to  which  vanadium  side  rods,  piston 


Art.  58.]  STEEL.  331 

rods,  and  crank  pins  are  submitted  is  the  same  as  that 
given  above  for  driving  axles. 

In  the  manufacture  of  locomotive  tires,  the  heat  treat- 
ment is  somewhat  different  from  that  set  forth  above,  as 
it  consists  of : 

"  (i)  In  reheating  the  tires  after  rolling,  and  then 
quenching  in  oil,  (2)  then  reheating  slowly  and  uniformly 
to  a  temperature  sufficiently  high  to  obtain  the  desired 
physical  properties.  The  tire  must  be  held  at  this  final 
temperature  at  least  two  hours,  which  is  considered  the 
minimum  time  required  for  the  changes  to  be  effected 
throughout  the  tire  section.  The  tire  should  then  be  with- 
drawn from  the  furnace  and  allowed  to  cool  in  still  air. 

1 '  The  recommended  temperature  for  quenching  is  about 
1600°  F.  The  final  heating  for  obtaining  the  physical 
properties  specified  should  be  approximately  noo  to 
1200°  F." 

It  is  obvious  that  material  with  such  physical  properties 
possesses  unusual  toughness  and  resilience.  For  that  reason 
it  is  specially  adapted  to  locomotive  springs  and  other 
similar  uses.  For  such  a  purpose  the  carbon  contained  is 
relatively  high.  Mr.  Norris  in  the  paper  already  indi- 
cated gives  the  following  as  a  suitable  chemical  composition : 

Chemical  Composition. 

Per  cent. 

Carbon 0.52  to  0.60 

Manganese 0.70  to  0.90 

Chromium 0.80  to  i i.o 

Vanadium Over  o.  16 

Phosphorus Not  over  0.04 

Sulphur Not  over  0.04 

This  material  requires  heat  treatment  consisting  of : 

"  (i)  Heating  and  quenching  in  oil,  (2)  then  reheating 


332 


TENSION. 


[Ch.  VII. 


or  drawing  back,  preferably  in  a  lead  bath,  and  cooling 
slowly.  The  time  in  the  lead  bath  should  be  10  to  15 
minutes. 

"The  recommended  temperature  for  quenching  is  from 
1575  to  1650°  F.  The  drawback  or  annealing  temperature 
should  be  approximately  from  900  to  1100°  F." 

When  such  material  is  tempered  for  railway  springs  it 
has  the  following  physical  properties : 


Elastic  limit,  Ibs.  per  sq.in 160,000-180,000 

Tensile  strength,  Ibs.  per  sq.in..  ..  190,000-230,000 

Elongation  in  2  inches 10-15% 

Reduction  of  area 30-45%  " 


This  material  possesses  the  highest  physical  properties 
of  the  steels  yet  used  for  commercial  purposes. 

Some  recent  tests,  June,  1915,  reported  by  the  American 
Vanadium  Company,  show  excellent  results  for  carbon- 
vanadium  steel  both  in  the  natural  condition  of  the  speci- 
mens and  after  simple  annealing  as  well  as  after  heat  treat- 
ment, the  latter  yielding  highest  results  generally,  but  not 
for  ultimate  resistance,  the  ductility,  however,  being  dis- 
tinctly lower  in  the  natural  condition.  The  following  table 
gives  the  results  of  the  tests  as  well  as  the  chemical  analysis 
and  treatment.  The  first  six  sets  of  values  belong  to  test 
specimens  taken  from  7 -inch  and  u-inch  axles,  while  the 
last  three  belong  to  specimens  from  connecting  rods. 


TESTS   OF   CARBON-VANADIUM    STEEL 


Carbon 

Manganese 

Phosphorus 

Sulphur 

Vanadium 

Chem.  Analysis.  . 

0-47% 

0.90% 

0.012% 

0.020% 

0.15% 

Art.  58.]     EFFECT  OF   HIGH  AND  LOW  TEMPERATURES. 
PHYSICAL  PROPERTIES 


333 


Treatment. 

Yield 
Point 
Ibs.  per 
sq.  in. 

Elastic 
Limit 
Ibs.  per 
sq.  in. 

Ultimate 
Resist. 
Ibs.  per 
sq.  in. 

Stretch 
in  2  in. 
Per- 
cent. 

Reduc- 
tion of 
Area 
Per- 
cent. 

Natural                  

7I.2OO 

68,OOO 

I23,OOO 

16.0 

3O.O 

Annealed  1450°  F  

56,OOO 

52,OOO 

9O,OOO 

24.0 

50.0 

O.  Q.  1600;  T.  1160°  F  

85,000 

82,OOO 

II2,5OO 

22.  O 

55.0 

Natural 

7S,OOO 

7O,OOO 

II7,OOO 

16  o 

28  5 

Annealed  1450°  F              

58,000 

54,OOO 

Q4,  OOO 

22    O 

4-7   O 

O  Q    1600-  T.  1160°  F  

87,000 

8O,OOO 

II5,OOO 

2O.  5 

S2    O 

Natural          

92,000 

85,000 

I3I,OOO 

17.0 

44.O 

Annealed  1450°  F 

7I.SOO 

67  ooo 

105  ooo 

2T,     S 

C2    o 

O  Q    1600-   T   1160°  F 

Q2.SOO 

86  ooo 

I23,OOO 

2O    ^ 

en   o 

Effect  of  Low  and  High  Temperatures  on  Steel. 

There  has  been  much  difference  of  opinion  expressed 
upon  the  effect  of  low  temperature  upon  steel,  especially 
upon  steel  rails.  The  high  number  of  breakages  in  steel 
rails  during  the  winter,  particularly  in  the  early  days  of  the 
use  of  steel  for  such  a  purpose,  has  given  the  impression 
that  low  temperatures  in  the  vicinity  of  zero  degrees  F., 
or  lower,  make  steel  brittle  and  hence  subject  to  sudden 
fracture  without  warning  in  the  cold  weather  of  winter. 
This  impression  has  been  shown  to  be  without  material 
foundation  in  rails  of  the  best  quality,  but  phosphorus 
makes  iron  and  steel  "  cold  short."  If,  therefore,  there 
should  be  a  sufficient  amount  of  phosphorus  present  steel 
or  iron  would  undoubtedly  become  more  liable  to  fracture 
at  low  temperatures.  In  the  early  days  of  rail  making, 
when  the  constituent  elements  were  not  so  carefully  con- 
trolled as  at  present,  it  is  highly  probable  if  not  practically 
certain  that  the  presence  of  phosphorus  accounted  for  many 
breakages  at  low  temperatures.  For  many  years,  howTever; 


334  TENSION.  [Ch.  VII. 

the  effects  of  the  prejudicial  hardeners  phosphorus  and 
sulphur  have  been  well  recognized  and  they  have  been  kept 
so  low  as  to  have  no  material  effect  upon  the  finished 
products. 

Again,  frozen  ground  in  the  winter  adds  somewhat  to 
the  rigidity  of  a  roadbed,  enhancing  to  some  extent  the 
effects  of  shocks  or  blows  to  which  rails  are  subjected  under 
rapidly  moving  heavy  train  loads.  Some  of  the  increased 
breakages  in  the  winter  are  probably  due  to  this  cause  and 
it  is  possible  that  a  great  majority  of  them  may  be  ac- 
counted for  in  this  way. 

On  the  whole  the  latest  experiences  do  not  seem  to 
indicate  that  with  the  excellent  quality  of  steel  now  pro- 
duced for  engineering  purposes  the  effects  of  low  tempera- 
tures are  at  all  serious,  but  that  they  may  be  ignored  when 
suitable  precautions  are  taken  in  the  processes  of  manu- 
facture. 

The  effect  of  high  temperature,  on  the  other  hand,  is  a 
matter  of  some  concern  in  connection  with  building  con- 
struction, since  the  ultimate  carrying  capacity  of  iron  or 
steel  may  be  seriously  affected  or  even  destroyed  by  the 
high  temperatures  of  conflagrations  unless  the  supporting 
members  are  protected  against  the  effects  of  intense 
heat. 

Figs.  3  and  4  represent  the  results  of  investigations  by 
Prof.  R.  C.  Carpenter,  formerly  of  Cornell  University,  who 
made  tensile  tests  on  wrought  iron  and  steel  circular  speci- 
mens .5  inch  in  diameter.  Fig.  3  is  self-explanatory.  It 
shows  the  graphical  relation  between  the  temperatures  of 
the  specimens  and  the  ultimate  tensile  resistance  per  square 
inch. 

•The  ductility  represented  by  the  final  elongations 
or  stretches  in  8  inches  at  the  corresponding  temperatures 
of  rupture  are  exhibited  in  Fig.  4. 


Art.  58.] 


STEEL. 


335 


Prof.  Carpenter  observes  "that  all  the  curves  have 
a  point  of  contraflexure  at  about  70°  F.,  and  another 
at  a  temperature  not  far  from  500°.  The  maximum 
strength  is  found  at  temperatures  of  400°  to  550°.  At 


LBS. 

150,000 


.140,000 


130,000 
120,000 

u  110,000 

< 

"  100,000 

I 

B  90,000 

& 

»  80,000 

H 

<  70,000 

=  co.ooo 


60,000  <i 


40,000 


80,000 


0   100   200   300   400   500   600   700   800   900 

TEMPERATURE  OF  SPECIMEN,  DEGREES  F. 


FIG.  3. 

temperatures  higher  than   this  all  the  materials   show  a 
rapidly  decreasing  strength." 

As  a  general  result  or  consensus  of  all  results,  includ- 
ing the  older  and  the  later,  it  may  be  stated  that  iron 
and  steel  lose  no  sensible  portion  of  their  resisting  capac- 
ity under  about  500°  Fahr.,  but  that  softening  is  liable 
to  begin  when  the  temperature  rises  much  above  that 
limit.  At  a  temperature  of  about  800°  Fahr.  these  metals 
may  lose  as  much  as  20  per  cent,  of  ultimate  resistance. 


336 


TENSION. 


[Ch.  VII. 


Hardening  and  Tempering. 

The  processes  of  hardening  and  tempering  are  not 
usually  applied  to  structural  steel,  but  to  those  higher 
grades  of  metal  used  for  such  special  purposes  as  tools 


1.75 


WROUGHT 


ROh 


7 


TOOL  S 


EEL 


0.75 


Q.25 


TOO 


LxST 


100° 


300 J  500° 

TEMPERATURE  OF   SPECIMEN,  DEGREES  F. 


700 c 


900 


FIG.  4 

or  wire.  The  hardening  process  consists  in  heating  the 
steel  to  vSuch  temperature  as  may  be  desired  to  accomplish 
a  given  purpose  and  then  quenching  in  water,  brine, 
oil,  molten  lead,  or  other  proper  bath.  The  temperature 
from  .which  the  quenching  is  done  may  be  that  indicated 


Art.  58.]  STEEL.  337 

by  an  orange  color;  it  depends  upon  the  size  or  character 
of  grain  of  metal  desired.  In  general  terms,  the  higher 
the  content  of  carbon,  the  more  marked  will  be  the  re- 
sults of  the  hardening  processes.  Quenching  has  a  com- 
paratively small  effect  upon  low  or  medium  structural 
steel. 

The  process  of  tempering  is,  in  reality,  supplementary 
to  the  process  of  hardening  in  the  manner  just  described. 
After  a  piece  of  steel  has  been  hardened  by  quenching 
so  that  its  temperature  is  that  of  the  air,  if  it  be  again 
heated  it  will  exhibit  different  colors  as  the  temperature 
is  increased.  The  first  noticeable  color  will  be  a  light 
delicate  straw,  then  deep  straw,  light  brown,  dark  brown, 
brownish  blue,  called  "  pigeon  wing,"  light  bluish,  light 
brilliant  blue,  dark  blue,  and  black,  after  which  the  temper 
is  completely  removed.  The  preceding  colors  are  due 
to  thin  films  of  oxide  that  form  on  the  exterior  surfaces 
of  the  pieces  as  the  temperature  increases.  When  this 
heating  is  stopped  at  any  color  and  'the  steel  allowed  to 
cool,  the  metal  is  said  to  be  drawn  to  the  temper  shown 
by  the  corresponding  color. 

The  tempers  at  different  colors  for  different  processes 
are  sometimes  stated  as  follows: 

Light  straw For  lathe-tools,  files,  etc. 

Straw "       "      .  "         "       " 

Light  brown "  taps,  reamers,  drills,  etc. 

Darker  brown 

Pigeon  wing "  axes,  hatchets,  and  some  tools. 

Light  blue "  springs. 

Dark  blue "  some  springs,  occasionally. 

Tempering  or  hardening  increases  both  the  elastic 
limit  and  ultimate  resistance,  but  decreases  the  ductility. 


338  TENSION,  [Ch.  VII. 

Annealing. 

The  processes  of  annealing,  like  those  of  hardening 
and  tempering,  produce  more  marked  results  in  the  higher 
steels  than  in  the  lower.  Steel  has  a  sensibly  varying 
density  at  different  temperatures;  in  other  words,  a 
given  weight  of  metal  will  occupy  sensibly  different  volumes 
at  different  temperatures.  Hence  if  a  piece  of  steel  be 
subjected  to  any  operation,  such  as  forging,  which  gives 
to  different  portions  concurrently  widely  varying  tem- 
peratures, those  portions  will  necessarily  be  subjected 
to  considerable  intensities  of  internal  stresses,  and  if 
those  stresses  are  not  removed  they  may  reduce  greatly 
both  the  ultimate  resistance  and  ductility.  In  the  higher 
grades  of  steel  and  in  special  steels  it  is,  therefore,  impera- 
tive to  anneal  members  which  have  been  subjected  to 
such  operations.  These  observations  are  specially  perti- 
nent to  such  high  steels  as  those  adapted  to  the  manufacture 
of  tools  or  other  similar  purposes.  In  general  it  is  neces- 
sary in  structural  engineering  practice  to  resort  to  anneal- 
ing only  in  the  case  of  eye -bars,  or  other  members  which 
have  been  subjected  to  the  operations  of  forging.  The 
process  consists  simply  in  heating  the  member  to  be 
annealed  to  about  a  cherry-red  temperature  until  the 
piece  is  heated  through,  and  then  allowing  it  to  cool  grad- 
ually to  a  normal  temperature.  At  the  cherry-red  heat 
the  metal  is  sufficiently  softened  to  allow  the  molecules 
to  readjust  their  relative  positions  so  as  to  remove  the 
internal  stresses.  After  the  operation  of  cooling  is  com- 
pleted the  metal  will  be  at  least  approximately,  if  not 
entirely,  in  a  condition  of  no  internal  stresses,  i.e.,  if  the 
annealing  has  been  properly  done.  The  more  gradually 
and  uniformly  the  cooling  is  accomplished  the  more  ex- 
cellent will  be  the  results.  Sometimes  resort  is  made  to 


Art.  58.]  STEEL.  339 

such  special  means  to  accomplish  these  ends  as  covering 
the  members,  after  bringing  them  to  a  proper  temperature, 
with  sand,  ashes,  or  other  similar  material,  to  insure  a 
slow  and  uniform  cooling. 

The  preceding  tables  show  what  is  always  found  in 
a  comparison  of  results  for  the  natural  and  the  an- 
nealed metal.  The  process  of  annealing  will  diminish 
the  ultimate  resistance  of  structural  steel  in  general  from 
about  4,000  to  6,000  or  8,000  pounds  per  square  inch, 
and  the  elastic  limit  will  be  reduced  correspondingly. 
These  effects  will  be  found  more  marked  as  the  metal  is 
finished  between  the  rolls  at  lower  temperatures.  In 
general,  steel  which  is  hardened  by  the  conditions  of 
manufacture,  like  that  of  comparatively  low  temperature 
in  rolling,  will  exhibit  greater  decreases  of  ultimate  resist- 
ance and  elastic  limit  under  annealing. 

The  process  of  annealing  increases  the  ductility  of 
the  steel,  since  it  softens  the  metal.  In  spite  of  the  re- 
duction in  ultimate  resistance  and  elastic  limit,  therefore, 
the  operation  gives  a  valuable  quality  to  the  steel. 

Effect  of  Manipulations  Common  to  Constructive  Processes; 
Also  Punched,  Drilled  and  Reamed  Holes. 

The  shop  treatment  of  steel  must  in  some  respects  be 
peculiar  to  that  metal  and  different  from  that  which 
characterizes  the  manufacture  of  wrought-iron  bridge  mem- 
bers. While  the  processes  of  punching  and  shearing  may 
not  be  specially  injurious  to  comparatively  thin  plates  and 
shapes  of  low  steel  and  of  the  lower  carbon  grades  of 
mild  steel  (perhaps  up  to  a  limit  of  65,000  pounds  per  square 
inch)  they  are  sufficiently  injurious  to  heavier  sections  and 
to  the  higher  grades  of  steel  to  necessitate  the  avoidance 
of  their  effects.  If  punches  and  dies  are  kept  in  good  sharp 
condition,  as  they  should  be,  the  prejudicial  effects  are 


340  TENSION.  [Ch.  VII. 

lessened.  The  effect  of  a  punch,  however,  under  the  best 
conditions  of  operation  is  not  to  make  a  smooth-sheared 
surface,  but  one  of  somewhat  ragged  or  serrated  character 
in  which  incipient  cracks  are  started  and  which  may  be 
continued  indefinitely  into  the  interior  of  the  metal  unless 
some  curative  procedure  is  employed. 

It  has  been  found  by  actual  test  that  the  region  affected 
by  the  punch  or  by  the  jaw  of  the  shear  extends  but  a 
short  distance  from  the  cutting-edge  of  the  tool.  Within 
that  region,  however,  the  metal  is  much  hardened  and  the 
loss  of  ductility  and  elevation  of  elastic  limit  is  due  to  that 
hardening.  The  decreased  ultimate  resistance  is  probably 
due  to  the  violent  disturbance  of  the  molecules  and  the 
resulting  minute  fissures  in  the  metal  within  the  same 
region.  In  riveted  work,  the  prejudicial  effect  is  therefore 
removed  by  reaming  the  punched  hole  to  a  diameter  about 
|  inch  larger  than  made  by  the  punch.  This  removes  a 
thin  ring  of  injured  metal  about  yg-  of  an  inch  thick,  and 
it  is  found  sufficient  for  the  purpose. 

In  large  and  heavy  work  it  has  come  to  be  the  practice 
by  the  best  shops  to  make  drilled  holes  in  which  cases  no 
question  of  the  injury  of  metal  can  arise.  The  use  of  the 
drill  leaves  a  sharp  edge  at  each  surface  of  the  plate  which 
tends  to' produce  a  shearing  effect  upon  the  corresponding 
rivet  sections.  Some  specifications  require  this  to  be  over- 
come by  a  quick  application  of  a  proper  tool  to  remove  the 
sharp  edge. 

The  general  effects  of  the  cutting  edge  of  the  shear 
are  precisely  the  same  as  those  of  the  punch,  as  the  opera- 
tion in  each  case  is  a  shear.  Hence,  if  sheared  edges 
are  planed  off  to  a  depth  of  one-sixteenth  to  one-eighth 
of  an  inch,  the  injured  metal  will  be  entirely  removed. 
The  hardening  effects  of  both  shearing  and  punching 
may  also  be  removed  by  the  process  of  annealing,  although 


Art.  58.]  STEEL.  341 

less  effectually  than  by  reaming  and  planing.  As  naturally 
would  be  inferred  by  experience  in  punching,  higher  steel 
and  thicker  plates  are  more  injuriously  affected  by  shear- 
ing than  low  steels  and  thinner  plates. 

In  consequence  of  the  irregular  edge  of  a  large  sheared 
plate,  bridge  specifications  frequently  require  that  at 
least  one-quarter  of  an  inch  of  metal  shall  be  removed 
from  the  edge  of  such  plates  by  planing. 

Steel  seems  to  be  very  sensitive  to  the  effects  of  hammer- 
ing or  working  at  what  is  termed  a  "  blue  heat."  Con- 
sequently it  is  necessary  to  heat  the  rivet  to  such  a  tem- 
perature as  will  enable  the  operation  of  heading  to  be 
completed  before  the  rivet  cools  to  the  blue  stage.  A 
bright  red  or  yellow  heat  is  requisite  for  good  work,  and 
the  rivet  should  be  held  under  a  pressure  of  fifty  or  sixty 
tons  per  square  inch  of  the  shaft  section  until  the  metal 
has  time  to  flow  throughout  the  rivet  length  and  thus 
completely  fill  the  hole,  otherwise  the  upsetting  will  be 
complete  at  and  in  the  vicinity  of  the  rivet-heads  only. 
An  additional  advantage  in  holding  the  rivet  under  the 
greatest  pressure  of  the  riveter  for  a  short  time  is  the 
fact  that  the  rivet  becomes  cool  enough  to  prevent  the 
separation  of  the  plates. 

The  forging  of  steel  requires  unusual  skill  and  ex- 
perience. When  a  piece  has  been  heated  to  a  proper 
temperature  it  should  be  kept  under  work  until  it  has 
fallen  in  temperature  to  a  proper  point  to  secure  all 
the  advantages  of  working,  but  of  course  not  below 
red  heat.  The  forging  should  be  done  with  a  hammer 
whose  weight  is  suitably  proportionate  to  the  mass  to  be 
forged.  If  the  hammer  is  too  light,  the  result  will  be  a 
surface  effect  only,  with  the  interior  but  little  changed. 
Pressure  forging,  with  appropriate  facilities  for  attaining  great 
pressures,  is  probably  capable  of  producing  the  best  results. 


342  TENSION.  [Ch.  VII. 

The  operation  of  annealing,  particularly  as  applied 
to  full-size  bars,  is  one  of  great  importance  in  the  manu- 
facture of  structural  steelwork.  The  metal  is  heated  as 
uniformly  as  possible,  so  that  undue  stresses  will  not  be 
developed,  to  a  bright  cherry-red,  corresponding  probably 
to  about  1 1  oo  or  1200  degrees  Fahr.,  and  then  allowed 
to  cool  gradually.  By  this  means  any  internal  stresses 
that  may  have  been  produced  by  the  process  of  forging, 
or  any  other  shop  manipulation,  are  eliminated.  The 
metal  is  sufficiently  softened  at  the  highest  temperature 
to  allow  the  molecules  to  adjust  themselves  to  a  condition 
of  essentially  no  stress,  and  if  the  cooling  is  gradual  the 
internal  stresses  will  not  be  re-developed. 

Change  of  Ultimate  Resistance,  Elastic  Limit  and  Modulus 
of  Elasticity  by  Rete sting. 

It  has  been  observed  from  the  earliest  experiences  in 
testing  steel  and  wrought  iron  that  if  a  piece  of  material 
be  subjected  to  an  intensity  of  tensile  stress  higher  than  the 
elastic  limit,  thus  producing  permanent  stretch,  the  ultimate 
resistance  will  be  materially  increased,  although  the  duc- 
tility is  generally  decreased.  Sufficient  investigation  has 
not  even  yet  been  undertaken  to  gage  the  full  significance 
of  such  phenomena,  but  enough  has  been  done  to  show 
some  important  results. 

It  is  yet  uncertain  whether  an  indefinitely  long  rest  may 
not  diminish  to  some  extent  at  least  the  enhanced  ultimate 
resistance  of  a  piece  of  metal  stressed  beyond  the  elastic 
limit.  Professor  Bauschinger  made  some  investigations  in 
this  special  field  many  years  ago  which  indicate  that  the 
elastic  limit  is  considerably  decreased  by  immediate  retest- 
ing,  but  that  such  a  decrease  does  not  take  place  if  a 
period  of  at  least  twenty -four  hours  or  possibly  more  elapses 
before  retesting.  Some  tests  indicate  that  the  elastic  limit 


JArt.  58.  STEEL.  343 

may  be  much  increased  even  by  suitable  periods  of  rest 
between  applications  of  loading. 

The  yield  point  appears  to  be  raised  materially  by  re- 
testing  and  the  same  observation  as  already  indicated  is 
equally  applicable  to  the  ultimate  resistance. 

Fracture  of  Steel. 

The  character  of  steel  fractures  has  been  incidentally 
noticed,  in  some  cases,  in  the  different  tables. 

If  the  steel  is  low,  or  if  some  of  the  higher  grades  are 
thoroughly  annealed,  the  fracture  is  fine  and  silky,  pro- 
vided the  breakage  is  produced  gradually.  In  other 
cases  the  fracture  is  partly  granular  and  partly  silky,  or 
wholly  granular. 

In  any  case  a  sudden  breakage  may  produce  a  granular 
fracture. 

The  Effects  of  Chemical  Elements  on  the  Physical  Qualities 

of.  Steel. 

Anything  more  than  a  meagre  statement  of  the  influ- 
ence«kOf  the  chemical  composition  of  steel  on  its  physical 
properties  is  obviously  out  of  place  here,  but  a  knowledge, 
however  slight  it  may  be,  of  the  influence  of  certain  ele- 
ments on  those  properties  is  so  essential  to  the  engineer 
in  his  structural  work  that  attention  should  at  least  be 
called  to  it. 

Although  other  elements  exert  highly  important  influ- 
ences upon  the  resisting  qualities  of  steel,  carbon  is  un- 
doubtedly the  most  prominent  hardener.  The  effect 
of  a  given  percentage  of  carbon,  at  least  within  certain 
rather  wide  limits,  is  to  give  greater  toughness  and  resist- 
ing qualities  to  steel  with  less  concurrent  brittleness  than 
any  other  contained  element.  It  is  made,  therefore, 
the  basis  of  classification  of  structural  steel,  the  low  steels 
being  low  in  carbon  and  the  high  steels  high  in  carbon. 


344  TENSION.  [Ch.  VII. 

The  metal  manganese  also  gives  to  steel  some  advan- 
tageous qualities.  At  the  present  time  it  seldom  enters 
steel  to  an  amount  less  than  .5  per  cent.,  nor  more  than 
about  i  per  cent.  Its  presence  seems  to  confer  the  capacity 
of  resisting  the  effects  of  high  temperatures  in  shop  pro- 
cesses. Metal  low  in  phosphorus  and  sulphur  appears 
to  require  less  manganese  than  that  which  is  higher  in 
those  impurities.  It  has  been  found  that  the  influence  of 
manganese  upon  steel  depends  in  a  rather  extraordinary 
manner  upon  its  amount.  If  the  content  reaches  1.5 
or  2  per  cent,  steel  becomes  practically  worthless  on 
account  of  its  brittleness,  but  when  a  content  of  6  or  7 
per  cent,  of  manganese  is  reached,  the  metal  becomes 
extremely  hard  and  acquires  to  a  high  degree  the  property 
of  toughness  by  quenching  in  water  without  becoming 
much  harder. 

When  steel  is  alloyed  with  more  than  about  7  per 
cent,  of  manganese,  manganese -steel  is  the  product,  wrhich, 
in  its  natural  state,  may  have  an  ultimate  tensile  resist- 
ance running  from  74,000  to  over  116,000  pounds  per 
square  inch.  When  quenched  in  water  the  ultimate 
tensile  resistance  of  the  same  metal  may  run  from  about 
90,000  pounds  per  square  inch  up  to  nearly  137,000  pounds 
per  square  inch.  Before  quenching  the  final  stretch 
ranged  from  i  to  4  or  5  per  cent.,  and  after  quenching 
frorri  4  to  44  per  cent.  The  preceding  figures  belong  to 
a  range  in  manganese  from  about  7  per  cent,  to  over  19 
per  cent,  concurrently  with  carbon  from  about  .61  per 
cent,  up  to  1.83  per  cent.  This  metal  is  an  interesting 
alloy,  but  is  never  used  in  structural  engineering  work. 

Opinions  vary  much  as  to  the  influence  of  silicon 
on  steel,  but  it  seems  now  to  be  well  established  that 
that  influence  within  the  limits  ordinarily  found  is  of 
minor  consequence,  or  at  least  not  prejudicial  to  either 


Art.  58.]  STEEL.  345 

resistance  or  ductility.  In  structural  steel  it  usually 
ranges  from  less  than  .03  to  .05  per  cent.,  while  in  rail 
steel  it  may  run  as  high  as  .3  per  cent.  Jn  some  excellent 
tool-steel  it  may  run  even  from  .2  to  .75  per  cent. 

Sulphur  is  an  impurity  carrying  with  it  highly  preju- 
dicial effects.  It  essentially  injures  metal  for  rolling,  as 
it  makes  the  steel  liable  to  crack  and  tear  at  the  usual 
temperatures  found  between  the  rolls.  It  also  diminishes 
capacity  to  weld.  Its  effects  may,  to  some  extent,  be 
overcome  by  the  presence  of  manganese  and  by  proper 
care  in  heating.  It  is,  however,  highly  prejudicial  as 
an  element  and  is  usually  kept  below  about  .04  per  cent. 

Of  all  the  objectionable  elements  found  in  steel,  phos- 
phorus has  the  position  of  primacy.  Although  'it  is  a 
hardener  which  may  increase  the  ultimate  resistance 
to  some  extent,  it  produces  brittleness  and  diminishes 
most  materially  the  capacity  to  resist  shock,  and  it  is 
one  of  the  chief  purposes  of  the  best  methods  of  steel 
production  to  reduce  phosphorus  to  the  lowest  practicable 
limit.  Its  effects  are  sometimes  erratic,  being  occasionally 
found  in  excess  in  apparently  good  material.  In  structural 
steel  it  is  seldom  permitted  to  run  over  .08  per  cent.,  and 
in  the  basic  processes  of  manufacture  it  frequently  falls 
to  .03  or  .04  per  cent. 

The  presence  of  .1  to  .25  per  cent,  copper  appears  to 
have  no  deleterious  effect  upon  steel  and  may  even  be 
beneficial.  As  high  as  i  per  cent,  of  copper  has  been 
found  in  steel  without  serious  effects  where  sulphur  was 
low. 

Aluminum  steel  is  an  alloy  containing  at  times  as 
high  as  5  to  6  per  cent,  of  aluminum.  The  effect  of  alumi- 
num on  ultimate  resistance  does  not  seem  to  be  prejudicial, 
nor,  again,  is  it  of  any  special  advantage;  nor  does  it 
act  seriously  upon  the  ductility  until  its  amount  approaches 


346  • 


TENSION. 


[Ch.  VII. 


about   2   per  cent,   or  more.     On  the  whole  it  does  not 
seem  to  be  a  valuable  element  for  steel. 

There  are  other  special  alloys  such  as  tungsten  and 
chromium  steel.  They  are  used  for  the  special  purposes 
of  tools  on  account  of  their  hardness,  which  is  so  extreme 
that  neither  quenching  nor  tempering  is  required.  They 
dp  not,  however,  enter  into  structural  use. 

Art.  59. — Copper,  Tin,  Aluminum,  and  Zinc,  and  their  Alloys- 
Alloys  of  Aluminum — Phosphor-bronze — Magnesium. 

Anything  like  a  complete  knowledge  of  the  physical 
properties  of  the  alloys  of  copper,  tin,  aluminum,  zinc,  etc., 
is  still  lacking,  although  many  investigations  have  been 
made  in  the  past  by  the  late  Prof,  R.  H.  Thurston  and 
others,  while  other  investigations  are  still  in  progress.  The 
character  of  many  of  these  alloys  changes  so  radically  for 
different  proportions  of  the  constituent  elements  and  under 
different  conditions  of  heat  and  other  treatment  that  the 
results  of  tests  are  as  varied  as  the  relative  amounts  of 
the  constituents  and  the  physical  conditions  which  attend 
the  tests.  Some  of  the  results  which  follow  belong  to  the 
earlier  work  of  Prof.  Thurston,  but  as  they  exhibit  the  same 
physical  qualities  as  the  corresponding  alloys  now  used  and 
as  the  later  investigations  do  not  cover  the  same  field,  they 
possess  real  value  and  are  retained. 

Table  I  gives  the  tensile  coefficients  of  elasticity  (E) 
of  copper  and  the  alloys  indicated  as  determined  by  Prof. 
Thurston. 

TABLE   I. 


Metal. 

Authority. 

E. 

Remarks. 

Gun-bronze.  .  .  . 
Alloy  
Alloy  
Tobin's  alloy.  .  . 
Copper.  . 

Thurston 
<  < 

11,468,000 
13,514,000 
14,286,000 

4,545,000 
9,091,000 

Copper,  0.90;   tin,  o.io  (nearly). 
Copper,  0.80;   zinc,  0.20. 
Copper,  0.625;   zinc,  0.375. 
Composition,  below  table. 
Cast  metal 

Art.  59-] 


COPPER,    TIN,  ALUMINUM,  ZINC,  ETC. 


347 


Tobin's  alloy  is  a  composition  of  copper,  tin,  and 
zinc,  in  the  proportions  (very  nearly)  of  58.2,  2.3,  and 
39.5,  respectively.  The  value  of  E  for  this  metal,  and 
those  for  the  two  preceding  and  one  following  it,  are 
calculated  for  small  stresses  and  strains  given  by  Prof. 
Thurston  in  the  "  Trans.  Am.  Soc.  Civ.  Engrs.,"  for  Sept., 
1881. 

There  will  also  be  found  in  Tables  VIII,  IX,  X  and 
XI  coefficients  of  elasticity  for  aluminum-zinc,  aluminum 
magnesium,  and  other  alloys,  and  for  magnesium,  alumi- 
num, and  zinc. 

TABLE  II. 

CAST   TIN. 


p- 

E. 

p. 

E. 

1,950 

i,  1  47,  ooo 

3,200 

96,400 

2,360 

472,000 

4,000 

4i,540 

2,580 

172,000 

Broke  at  4,200  Ibs. 

TABLE  III. 
CAST    COPPER. 


p- 

E. 

P' 

E. 

800 

10,000,000 

12,000 

18,750,000 

2,000 

9,091,000 

13,600 

8,193,000 

4,000 

9,091,000 

16,000 

2,235,000 

8,000 

14,815,000 

22,000 

137,000 

Broke  at  29,200  Ibs. 

The  values  of  E  (stress  over  strain)  for  different  inten- 
sities of  stress  (pounds  per  square  inch)  for  cast  tin,  cast 
copper,  and  Tobin's  alloy,  are  given  in  Tables  II,  III, 
and  IV. 


348 


TENSION. 


[Ch.  VII. 


"  p"  is  the  intensity  of  stress  in  pounds  per  square  inch, 
at  which  the  ratio  E  exists. 

Each  of  these  metals  is  seen  to  give  a  very  irregular 
elastic  behavior. 

Tables  II,  III,  and  IV  are  computed  from  data  given 
by  Prof.  Thurston  in  the  United  States  Report  (page 
42,5)  and  "  Trans.  Am.  Soc.  Civ.  Engrs.,"  already  cited. 

TABLE  IV. 

TOBIN'S  ALLOY. 


p. 

E. 

P- 

E. 

2,000 

4,545,000 

18,000 

5,455,000 

4,000 

4,545,000 

24,000 

5,941,000 

6,000 

4,088,000 

30,000 

6,250,000 

8,000 

4,938,000 

40,000 

6,390,000 

10,000 

5,263,000 

50,000 

4,744,000 

14,000 

5,110,000 

60,000 

3,436,000 

Broke  at  67,600  Ibs. 

Ultimate  Resistance  and  Elastic  Limit. 

Table  V  is  abstracted  from  the  results  of  the  experi- 
ments of  Prof.  Thurston  as  given  in  the  "  Report  of  the 
U.  S.*  Board  Appointed  to  Test  Iron,  Steel,  and  other 
Metals,"  and  "  Trans.  Am.  Soc.  of  Civ.  Engrs.,"  Sept., 
1 88 1.  The  composition  of  the  various  alloys  was  as 
given  in  the  table,  which  also  contains  results  for  pure 
copper,  tin,  and  zinc.  All  the  specimens  were  of  cast 
metal. 

The  mechanical  properties  of  the  copper-tin-zinc  alloys 
have  been  very  thoroughly  investigated  by  Prof.  Thurston 
("Trans.  Am.  Soc.  of  Civ.  Engrs.,"  Jan.  and  Sept.,  1881). 
As  results  of  his  work  he  has  found  that  the  ultimate 


Art.  59.]  COPPER,  TIN,  ALUMINUM,  ZINC,  ETC. 

TABLE  V. 


349 


Percentage  of 

Pounds  Stress  per 
Square  Inch  at 

Per  Cent.,  Fina- 

Copper. 

Tin. 

Zinc. 

Elastic 
Limit. 

Ultimate 
Resistance. 

Stretch. 

Contrac- 
tion. 

TOO 
IOO 
IOO 
90 
80 
70 
62 
52 

39 
29 

21 
IO 
OO 

OO 
OO 

Gun 
90 
80 
62.5 
58.2 

IOO 

90.56 
81.91 

71  .  20 

60.94 

58.49 

49.66 
41.30 
32.94 

20.  8l 

10.30 

0.0 

70.0 

57.50 

45-0 
'66.25 
58.22 
10.00 
60.00 
65.00 

OO 
OO 
OO 
IO 
20 
30 

38 
48 

61 
7i 
79 
90 

IOO 

Queens!'  d 

IOO 

Banca 

IOO 

Bronze 

IO 
OO 
00 

2-3 

0.0 
0.0 
0.0 
0.0 
0.0 
0.0 
0.0 
0.0 

o.o 

0.0 
0.0 

o.o 
8;  75 

21  .25 
23-75 
23-75 
2.30 
50.00 
IO.OO 

20.00 

'  00 
00 
00 
00 
00 
00 
00 
OO 
OO 
OO 
OO 
OO 
OO 

OO 
OO 
OO 

20 

37-5 

39.5 

o.o 

9.42 
17.99 
28.54 
38.65 

41  .  10 

50.14 
58.12 
66.23 
77.63 

88.88 

100.00 

20.25 
21.25 
31.25 

10.00 

39.48 

40.00 
30.00 
15.00 

11,620 
11,000 

14,400 

15,740 

19,872 
12,760 
27,800 
26,860 
32,980 

5,585 
688 

2,555 
2,820 
1,648 
4,337 
6,450 
3,500 

2,760 
3,5oo 

31,000 
33,HO 
48,760 
67,600 
29,200 

0.05 
0.005 
0.065 
0.037 
0.004 

IO.O 

8.0 
15-0 

13.5 
oo.o 
oo.o 
oo.o 
oo.o 
oo.o 
oo.o 
oo.o 

15,0 

75-0 
47-0 
86.0 

40.0 

29-5 
8.0 
16.0 

5,585 

688 

2,555 
2,820 

0.07 
0.36 

3,500 
1,670 

2,000 
10,000 

0.36 

4.6 

32.4 
31-0 

4-0 

7-5 

10,000 
9,000 
16,470 
27,240 
16,890 
3,727 
i,774 
9,000 
14,450 
4,050 
18,000  (?) 
1,300 
2,196 

3,294 
30,000  (?) 
5,ooo  (?) 
21,780  (?) 

32,670 
30,510 
41,065 
50,450 
30,990 
3,727 
i,774 
9,000 
14,450 
5,400 
31,600 
1,300 
2,196 

3,294 
66,500 
9,300 
21,780 
3,765 

31-4 
29.2 
20.7 

10.  I 

5-0 

43-0 
38.0 
28.0 
17.0 
ii.  5 

o.o 

0.0 
0.0 
0.0 

0.16 

0-39 
0.69 
0.36 

3-13 
0.7 

0.15 

7.0 

o.o 
o.o 

35° 


TENSION. 
TABLE  V. — Continued. 


[Ch.  VII. 


Percentage  of 

Pounds  Stress  per 
Square  Inch  at 

Per  Cent.,  Final 

Copper. 

Tin. 

Zinc. 

Elastic 
Limit. 

Ultimate 
Resistance. 

Stretch. 

Contrac- 
tion. 

yo.OO 

IO.OO 

2O.OO 

24,000(?) 

33,!40 

0.31 



75.00 

5.00 

2O.OO 

I2,OOO(?) 

34,960 

3-2 

5-4 

80.00 

10.00 

IO.OO 

I2,OOO(?) 

32,830 

1.6 

4.0 

55-00 

0.50 

44-50 

22,OOO 

68,900 

9-4 

25.0 

60.00 

2.50 

37-50 

22,OOO 

57,400 

4-9 

6.6 

72.50 

7.50 

2  .OO 

II,OOO 

32,700 

3-7 

ii  .0 

77-50 

12.5 

IO.OO 

2O,OOO 

36,000 

0.7 

0.0 

85.00 

12.5 

2.50 

I2,OOO(?) 

34,500 

1-3 

3-0 

The  values  of  the  elastic  limit  in  the  lower  part  of  the  table  were  not  at 
all  well  denned. 

tensile  resistance,  in  pounds  per  square  inch,  of  "  ordinary 

bronze,  composed  of  copper  and  tin, as  cast  in  the 

ordinary  course    of   a   brassfounder's  business,"  may    be 
well  represented  by 

Tc  =  30,000  -f  i  ,ooo£ ; 

"  where  t  is  the  percentage  of  tin  and  not  above  15  per 
cent." 

"  For  brass  (copper  and  zinc)  the  tenacity  may  be 
taken  as: 

72  =  30,000 +  500.2, 

where  z  is  the  percentage  of  zinc  and  not  above  50  per 
cent." 

He  found  that  a  large  portion  of  the  copper-tin-zinc 
alloys  is  worthless  to  the  engineer,  while  the  other,  or 
valuable  portion,  may  be  considered  to  possess  a  tenacity, 
in  pounds  per  square  inch,  well  represented  by  combining 
the  above  formulae  as  follows: 

Tzf  =  30,000  +  i  ,oooZ  -f-  5002. 
These  formulae  are  not  intended  to  be  exact,  but  to 


Art.  59.]  COPPER,  TIN,  ALUMINUM,  ZINC,  ETC.  351 

give  safe  results  for  ordinary  use  within  the  limits  of  the 
circumstances  on  which  they  are  based. 

Prof.  Thurston  found  the  "  strongest  of  the  bronzes" 
to  be  composed  of: 


Copper 55.0 


100.00 


This  alloy  possessed  an  ultimate  tensile  resistance  of 
68,900  pounds  per  square  inch  of  original  section,  an 
elongation  of  47  to  51  per  cent,  and  a  final  contraction 
of  fractured  section  of  47  to  52  per  cent. 

The  first  and  sixth  alloys  of  copper,  tin,  and  zinc,  in 
Table  V,  are  called  by  Professor  Thurston  "Tobin's  alloy." 
"  This  alloy,  like  the  maximum  metal,  was  capable  of 
being  forged  or  rolled  at  a  low  red  heat  or  worked  cold. 
Rolled  hot,  its  tenacity  rose  to  79,000  pounds,  and  when 
moderately  and  carefully  rolled,  to  104,000  pounds.  It 
could  be  bent  double  either  hot  or  cold,  and  was  found 
to  make  excellent  bolts  and  nuts." 

As  just  indicated  for  the  particular  case  of  the  Tobin 
alloy,  the  manner  of  treating  and  working  these  alloys 
exerts  great  influence  on  the  tenacity  and  ductility. 

Professor  Thurston  states:  "brass,  containing  copper 
62  to  70,  zinc  38  to  30,  attains  a  strength  in  the  wire  mill 
of  90,000  pounds  per  square  inch,  and  sometimes  of  100,000 
pounds." 

All  of  Professor  Thurston's  specimens  were  what  may 
be  called  "long"  ones,  i.e.,  they  were  turned  down  to 
a  diameter  of  0.798  inch  for  a  length  of  five  inches,  giving 
an  area  of  cross-section  of  0.5  square  inch. 


3S2 


TENSION. 


Alloys  of  Aluminum. 

Prof.  R.  C.  Carpenter,  of  Cornell  University,  in  the 
transactions  of  the  Am.  Soc.  Mech.  Engrs.,  vol.  xix,  has 
reported  a  number  of  interesting  and  valuable  tests  of 
alloys  of  aluminum,  as  well  as  tests  of  pure  magnesium. 

TABLE  VL 
ALLOYS  OF  GREATEST  RESISTANCE. 


Percentage  of 

Ultimate 
Resistance, 
Lbs.  per 

Specific 
Gravity. 

Per  Cent,  of 
Final  Stretch. 

Aluminum. 

Copper. 

Tin. 

Square  Inch. 

85- 

7-5 

7-5' 

30,000 

3.02 

4- 

6.25 

87.5 

6.25 

63,000 

7-35 

3-8 

5- 

5- 

90. 

II,OOO 

6.82 

IO.  I 

The  greater  part  of  the  results  for  the  aluminum-tin- 
copper  alloys  are  given  in  Table  VII,  but  the  composition 
of  those  giving  the  greatest  ultimate  resistances  are  ex- 
hibited in  Table  VI.  It  will  be  observed  that  the  highest 
ultimate  resistance  belongs  to  the  alloy  of  greatest  density 
but  the  alloy  of  least  resistance  has  nearly  as  great  density. 
The  ductility  of  none  of  these  alloys  of  greatest  ultimate 
resistance  is  specially  marked;  indeed,  the  ductility  is 
very  low  except  in  the  case  of  the  least  ultimate  resistance. 

The  composition  and  corresponding  elastic  limits  and 
ultimate  resistances  of  aluminum-tin-copper  alloys  will 
be  found  in  Table  VII.  Like  all  the  aluminum  alloys 
the  specific  gravity  varies  between  wide  limits,  being 
low  where  there  is  much  aluminum  and  high  where  there 
is  little.  The  ductility  is  low  in  all  cases  except  in  that 
of  pure  tin  or  the  alloy  in  which  it  appears  to  the  extent 
of  90  per  cent.  There  is  in  this  table  the  usual  wide  range 
of  physical  qualities  belonging  to  such  a  series  of  mixtures. 


Art.  59.] 


COPPER,  TIN,  ALUMINUM,  ZINC,   ETC. 


353 


TABLE    VII. 
ALUMINUM  ALLOYS. 


Composition,  Per  Cent, 
by  Weight. 

Ultimate  Resistance, 
Lbs.  per  Square  Inch, 

Elastic  Limit, 
Lbs.  per 
Square  Inch. 

Specific 
Gravity. 

Final 
Stretct- 
PerCen 
in  6 
Inches, 

Al. 

Tni 

Cu. 

A. 

B. 

IOO 

90 

80 
60 
40 

20 

27,000 
40,815 
32,209 
1,966 
849 
4,800    • 
15,000 
15,476 
18,580 
4.416 

915 
2,221 

3,505 
11,582 

5,999 
1,198 

993 

3,798 

28,330 
42,038 
34,200 
2,225 
1,077 
5,672 
I4,3i6 
17,070 
21,140 
5,950 
1,123 
2,622 

3,933 
10,418 
5,922 

1,200 
96l 

3,997 

I2,OOO 

13,832 
24,829 
* 
* 
* 

6,432 
8,227 

13.329 

* 
* 

6-5 
7-6 
6.5 
5-7 
5-05 
4.91 
2.67 
2.82 
3-09 
3-53 
4-4 

5-21 

7-3 
6.77 
6.24 

5-55 
4.96 

4.48 

6.5 
4.0 
0.8 

5.6 
3- 

1.2 

•3 

35-51 
10.  15 
I.I 

• 

5 
10 

20 

30 

40 

100 

90 

80 
60 
40 

20 

5 

10 
20 
30 
40 

5 

10 
20 
30 
40 
100 

90 

80 
60 
40 

20 

5 

10 

20 

3° 
40 

5 

IO 

20 

30 
40 

5 

10 
20 

30 
40 

4,823 
2,988 
* 
* 
* 

A.  Results  of  first  melting.     B.  Results  of  second  melting. 
Test  pieces  6  in.  between  shoulders,  diam.  \  inch. 
*  Could  not  be  turned  in  the  lathe. 

The  results  in  this  table  were  obtained  by  Messrs.  Geb- 
hardt  and  Ward,  at  the  testing  laboratory  of  Sibley  Col- 
lege of  Mechanical  Engineering,  Cornell  University,  in 
1896. 

The  physical  properties  of  aluminum-zinc  alloys,  in- 
cluding those  metals  unalloyed,  are  equally  fully  given 
in  Table  VIII,  as  well  as  the  values  of  the  coefficients  of 
elasticity.  There  is  not  as  wide  variation  of  results  in  this 
table  as  in  Table  VII,  although  there  is  a  considerable 
range  of  ultimate  resistance,  especially  if  the  results  for 
unalloyed  zinc  be  included.  It  will  be  observed  that 
this  table  also  includes  the  intensity  of  stress  found  in 


354 


TENSION. 


[Ch.  VII. 


TABLE   VIII. 
ALUMINUM-ZINC  ALLOYS. 


Percentage. 

Specific 
Gravity. 

Ultimate 
Resist- 
ance, 
Lbs.  per 
Sq.  in. 

Transverse 
Tests. 
Maximum 
Fibre 
Stress 
Lbs.  per 
Sq.  In. 

Coefficient 
of 
Elasticity. 

Remarks. 

Alumi- 
num. 

Zinc. 

100 
100 

90 
90 

85 
85 
80 
80 

75 
75 
70 
70 
66§ 

65 
60 
60 
50 
50 
25 
25 

0 

O 
o 

IO 
IO 

15 
15 

20 
20 

25 
25 
30 
30 

33 
35 
40 
40 
50 
50 
75 
75 

100 

2.67 
2.67 
2.77 
2.74 
2.918 
2.918 
2.998 
2-975 
3.15 
3.14 
3.I9I 
3.24 

3.326 
3.471 
3-57 

14,460 
16,750 
17,940 

14,500 
14,150 
18,950 

28,091 

6,535,ooo 

Shrinkage  uneven. 

Shrinkage  uneven 
«               « 

«              « 
Shrinkage  even. 

Poor  specimen. 

(  Elongation  of  all  the 
•<    specimens  less  than 
(    i  per  cent. 

7,710,000 
9,260,000 

18,100 
21,850 

22,940 

24,400 
23,950 

9,110,000 

34,600 

45,080 
43,200 
41,200 

8,210,000 

8,178,000 

19,770 
19,300 
19,060 
13,175 
I4»I5° 

2,522 

40,350 
38,100 
39,850 
25,500 

7,556 

8,540,000 
8,500,000 
8,670,000 
6,680,000 

7.19 

NOTE. — The  experimental  results  given  in  Table  IX  are  those  of  Messrs. 
Hunt  and  Andrews,  obtained  at  Sibley  College  of  Mechanical  Engineering, 
Cornell  University,  in  1894. 

TABLE    IX. 
TENSILE  TESTS  OF  MAGNESIUM— CAST  METAL. 


Number  of 
Test  Piece. 

Diameter. 

Ultimate 
Resistance, 
Lbs.  per 
Square  Inch. 

Elastic  Limit, 
Lbs.  per 
Square  Inch. 

Final 
Extension, 
Per  Cent. 

Coefficient  of 
Elasticity. 

I    

•433 

23,800 

8,800 

4-2 

2,040,000 

2 

AT.T. 

22  050 

i,  860  ooo 

•?    . 

.442 

2O,9OO 

10,780 

1.8 

2,060,000 

A 

435 

I9,5OO 

8,400 

2.5 

1,830,000 

X      , 

.424 

24,8OO 

7,090 

3.  I 

1,930,000 

6 

AT.2 

22   5OO 

2    ^ 

Art.  59-1 


COPPER,   TIN,  ALUMINUM,  ZINC,  ETC. 


355 


TABLE  X. 

ALLOYS  OF  ALUMINUM  AND  MAGNESIUM 


Number  of 
Test  Piece. 

Percentage  of 
Magnesium. 

Specific 
Gravity. 

Ultimate 
Resistance, 
Lbs.  per 
Square  Inch. 

Elastic  Limit, 
Lbs.  per 
Square  Inch. 

Coefficient  of 
Elasticity. 

I    

o 

2.67 

13,685 

4,900 

1,690,000 

2    

2 

2.62 

15,440 

8,700 

2,650,000 

•i    . 

K 

2.  59 

17,850 

13,090 

2,917,000 

A 

IO 

2  .55 

19,680 

14,600 

2,650,000 

3O 

2.29 

5,OOO 

the  extreme  fibres  of  beams  subjected  to  transverse  load- 
ing. Although  these  values  are  not  required  at  this 
point,  it  is  more  convenient  to  insert  them  here  and  refer 
to  them  in  the  article  devoted  to  the  flexure  of  such  beams. 
The  sizes  of  the  specimens  subjected  to  transverse  load- 
ing are  not  given,  but  they  were  small. 

TABLE  XI. 


Character  of  Alloys. 


Test  Piece. 

Composition  and  Remarks. 

Tension. 

Transverse. 

Al. 

Cu. 

Tin. 

Elastic. 

Ulti- 
mate. 

Elastic. 

Ulti- 
mate. 

3 

I 

2 

3 

% 
100 

93 
75-7 

% 

% 

4,OOO 
6,250 

12,055 

18,555 
35,075 

2,345 
9,000 

7 
3 

25,250 
23,420 

20%  zinc,  1.3  man. 

Rolled. 

4 

6 

7 
8 

9 

10 

100 
100 

98 
98 
96 
96 
96.5 

i  inch  bars.  
1     ' 

12,500 

17,185 

I7,i54 
18,870 
13,720 
18,870 
22,300 
30,880 
26,313 

2 

2 

4 
4 

2 



i                       

3         ' 
4                                 

I                                    .... 

9,OOO 
l6,OOO 

18,647 
23,045 



3       '                       

\\%  chromium. 

19,000 

26,310 

«! 

ii 

12 
13 
14 
15 

98 
96 

94 
92 
90 

2 

2,150 
2,400 
2,250 
2,OOO 
1,750 

8,622 
9,565 
9,315 
7,270 

7,352 

4. 

— 

6 

8 

10 

Resistance,  Pounds  per  Square  Inch. 


356 


TENSION. 


[Ch.  VII. 


The  experimental  results  given  in  Tables  IX  and  X 
were  also  established  at  the  testing  laboratory  of  Sibley 
College  of  Mechanical  Engineering  of  Cornell  University. 
The  tests  were  made  by  Messrs.  Marks  and  Barraclough, 
graduate  students  in  1893.  Table  IX  gives  results  for 
pure  magnesium,  including  the  coefficients  of  elasticity 
and  the  final  stretch,  while  Table  X  exhibits  the  results 
for  alloys  of  aluminum  and  magnesium,  the  per  cent. 
of  magnesium  being  shown  in  one  of  the  columns,  the 
remaining  per  cent,  being  aluminum.  The  ultimate  resist- 
ances given  in  Table  IX  show  that  magnesium  is  a  metal 
of  considerable  tensile  resistance,  especially  in  comparison 
with  its  density,  its  specific  gravity  being  but  1.74,  that 
of  aluminum  being  2.67. 

Table    XI    exhibits    the    elastic    limits    and    ultimate 

TABLE   XL — Continued. 


Final  Stretch 
Per  Cent. 

Final 
Contraction  , 
Per  Cent 

Hardness 

Specific  Gravity 
of  Specimen. 

Coefficient  of  Elasticity, 
Lbs.  per  Square  Inch. 

(Tension 

(Tension 

(Relative). 

Pieces). 

Ten- 

Trans- 

Tension. 

Transverse. 

sion. 

verse. 

5-62 

jo.  93 

3.6l 

2.670 

2-654 

8,385,000 

8,440,000 

1.  00 

3.08 

12.87 

2  .  830 

2.810 

11,115,000 

8,065,000 

•15 

i.77 

35.56 

3-I][7 

3-055 

9,685,000 

8,o6o,OOO 

8.49 

38-30 

7.12 

2.710 

9,780,000 

10,110,000 

6  .94 

2    71  S 

IO  OOO  OOO 

19.49 

39.02 

6.79 

2.725 

9,505,000 

10,330,000 

12  .  3O 

2    756 

Q  6OO  OOO 

3-62 

10.  10 

12.42 
I  T.     -5C 

2.774 

2    777 

10,440,000 

10,595,000 
10  070  ooo 

I-3I 

9.78 

14.09 

2-759 

9,850,000 

9,813,000 

4.00 

8.64 

3-71 

2.689 

5,435,000 

5.38 

6.86 

3-74 

2-739 

6,210,000 

5-19 

7-97 

3-49 

2.771 

5,035,000 

3-06 

5-41 

3-33 

2.804 

5,175,000 

3.87 

8.89 

3-09 

2.856 

6,675,000 

Art.  59.] 


COPPER    TIN,  ALUMINUM,  ZINC,  ETC. 


357 


resistances  of  all  the  different  alloys  shown  in  the  table,  and 
in  the  conditions  also  exhibited  by  the  table,  i.e.,  whether 
cast  or  rolled.  There  are  also  given  coefficients  of  elasticity 
for  both  tension  and  transverse  tests,  as  well  as  elastic 
limits  and  ultimate  stresses  (intensities)  in  the  extreme 
fibres  of  small  beams,  to  which  reference  will  be  made 
in  the  article  devoted  to  transverse  resistance. 

It  will  be  observed  that  both  the  elastic  limits  and 
the  ultimate  resistances  of  Table  XI  are  found  within 
the  range  exhibited  by  the  results  already  shown  in  the 
preceding  tables. 

If  desired,  diagrams  can  readily  be  constructed  from 
the  results  of  each  table  which  will  show  the  variations 
of  physical  quantities  corresponding  to  the  variations  of 
composition  of  the  alloys. 

In  1895  the  Fairbanks  Company  tested  at  their  New 
York  office  four  specimens  of  Tobin  bronze  manufactured 
by  the  Ansonia  Brass  and  Copper  Co.,  with  the  following 
results. 

ROLLED  TOBIN  BRONZE  PLATES— SPECIMENS   8  INCHES  LONG. 


Specimen, 
Inches. 

Resistance  in  Pounds  per  Sq.  Inch. 

Per  Cent.,  Final 

Elastic. 

Ultimate. 

Stretch. 

Contraction. 

iX.iSs 

iX.'25 
iX.25 

51,350 
51,350 
56,000 
56,450 

78,920 
78,810 
79,200 
79,640 

20.5 
17-5 
17.5 
16.25 

45-4 
44-33 
43-2 
40.72 

Alloys  of  Aluminum  and  Copper. 

In  1907,  Prof.  H.  C.  H.  Carpenter,  M.A.,  Ph.D.,  and 
Mr.  C.  A.  Edwards,  made  their  Eighth  Report  on  alloys 
of  aluminum  and  copper  to  the  Alloys  Research  Committee 
of  the  Institution  of  Mechanical  Engineers  of  Great  Britain. 


358 


TENSION. 


[Ch.  VII. 


This  alloy  is  known  as  "  aluminum  bronze  "  or  "  gold." 
These  investigators  made  over  a  thousand  tests  in  tension 
and  torsion  and  in  other  ways,  including  heat  treatment  for 
both  cast  and  rolled  material.  The  investigation  is  one  of  the 
most  important  ever  made  with  this  class  of  alloys.  Out 
of  the  great  number  of  tests  contained  in  the  report,  Table 
XII  has  been  selected  as  sufficiently  typical  for  the  purpose 
of  conveying  a  correct  impression  of  the  character  of  the 
work  done. 

TABLE   XII. 

The  percentage  of  aluminum  only  is  given  in  the  Table,  as  the  alloy  is  of 
aluminum  and  copper,  the  remaining  percentage  being  copper. 


No. 

Al. 
per  cent. 

Yield  Point 
Ibs.  per  sq. 
in. 

Ult.  Resist. 
Ibs.  per  sq. 
in. 

Elastic 
ratio. 

Elongation 
in  2  inches 
per  cent. 

I 

O.I 

8,512 

25,760 

•33 

46 

2 

I.  O6 

6,720 

30,O20 

.22 

52 

3 

2.1 

7,616 

30,240 

•25 

53-5 

4 

2.99 

8,512 

32,480 

.26 

60 

5 

4-05 

7,840 

37,410 

.21 

83 

6 

5-oy 

9.632 

40,540 

.24 

75 

7 

5-76 

10,752 

39,870 

•27 

67 

8 

6-73 

10,752 

41,780 

.26 

9 

7-35    ' 

14,784 

47,710 

•31 

7i 

10 

8.12 

17,248 

55,800 

•31 

58 

ii 

8.67 

21,952 

62,944 

•35 

48 

12 

9-38 

21,728 

68,050 

•32 

36.2 

13 

9-9 

25,312 

71,010 

•36 

21.7 

14 

10.78 

3L584 

59,750 

.48 

9.0 

15 

H-73 

3i,36o 

56,960 

•55 

5 

16 

13.02 

44,240 

44,240 

i 

It  will  be  observed  that  the  specimens  were  of  cast 
metal.  While  the  rolled  specimens  give  somewhat  higher 
ductility,  in  the  main  there  is  much  less  difference  than 
would  probably  have  been  anticipated.  Although  the 
elastic  ratio,  i.e.,  the  ratio  of  the  elastic  limit  over  the 
ultimate,  is  somewhat  higher  for  the  rolled  specimens,  the 
difference  on  the  whole  is  not  great,  except  in  a  compara- 


Art.  59.]  COPPER,   TIN,  ALUMINUM,  ZINC,  ETC.  359 

tively  few  instances.  In  fact,  the  differences  in  results 
found  by  the  investigators  between  the  cast  and  rolled 
metal  are  much  smaller  than  might  have  been  expected. 

The  authors  of  the  report  state,  among  other  obser- 
vations : 

"  (a)  The  limit  of  industrially  serviceable  alloys  must 
be  placed  at  1 1  per  cent,  of  aluminum!  For  most  purposes 
the  limit  might  be  put  at  10  per  cent.,  beyond  which  there 
is  a  rapid  fall  of  ductility  with  no  rise  of  ultimate  resist- 
ance. .  .  . 

"  (b)  Between  these  limits  the  alloys  fall  into  two 
classes:  i.  those  containing  from  o  to  7.35  per  cent,  of 
aluminum:  2.  Those  containing  from  8  to  n  per  cent,  of 
aluminum.  Class  i  represents  material  of  apparently  low 
yield  point  and  moderate  ultimate  stress,  but  of  very  good 
ductility.  The  introduction  and  further  addition  of  alumi- 
num causes  a  gradual  increase  of  strength  but  hardly  affects 
the  ductility.  It  is  true  that  as  regards  the  steadiness  of 
the  ductility  this  has  only  been  established  for  the  rolled 
bars.  But  the  sand  and  chill  castings  have  shown  the  same 
kind  of  variations  as  the  rolled  bars  in  all  the  properties 
examined.  .  .  . 

"  Into  Class  2  come  alloys  of  relatively  low  yield  point 
but  good  ultimate  stress.  From  8  to  10  per  cent,  of  alumi- 
num the  ductility  is  also  good.  ..." 

To  gain  an  adequate  idea  of  the  physical  properties  of 
the  various  grades  of  this  alloy  of  aluminum  and  copper 
requires  a  full  scrutiny  of  the  entire  report. 

Bronzes  and  Brass  Used  by  the  Board  of  Water  Supply  of 
New  York  City. 

In  the  construction  of  the  Additional  Catskill  Water 
Supply  for  the  city  of  New  York  by  the  Board  of  Water  Sup- 
ply a  large  amount  of  bronze  castings  and  rolled  bronze,  as 


360  TENSION.  [Ch.  VII. 

well  as  brass,  was  used  for  a  great  variety  of  large  and 
small  articles  varying  from  a  number  of  tons  in  weight  each 
to  a"  few  pounds,  such  as  small  bolts.  The  specifications 
prescribed  that  "  Whenever  the  term  '  bronze  '  is  used  in 
these  Specifications  in  a  general  way  or  on  the  drawings, 
without  qualification,  it  shall  mean  manganese  or  vanadium 
bronze  or  monel  metal.  .  .  . 

"  The   minimum   physical    properties    of  bronze  shall, 
except  as  otherwise  specified,  be  as  follows : 
Castings: 

Ultimate  tensile  strength 65,000  Ibs.  per  sq.in. 

Yield  point 32,000  Ibs.  per  sq.in. 

Elongation 25  per  cent. 

Rolled  Material: 

Ultimate  strength 72,000  Ibs.  per  sq.in. 

Yield  point 36,000  Ibs.  per  sq.in. 

Elongation 28  per  cent. 

Rolled  material,  thickness  above  one  inch: 

Ultimate  strength 70,000  Ibs.  per  sq.in. 

Yield  point 35,ooo  Ibs.  per  sq.in. 

Elongation 28  per  cent." 

The  modulus  of  elasticity  E  for  tension  and  compression 
was  about  14,000,000. 

The  requirements  of  these  specifications  were  even 
exceeded  both  in  resistances  and  in  ductility.  Much  trouble, 
however,  was  experienced  by  the  rolled  metal  exhibiting 
cracks  and  failures  in  articles  large  and  small,  in  many  cases 
even  before  put  in  place  in  the  work  and  subjected  to  duty. 
Such  difficulties,  however,  were  not  experienced  in  castings. 
Investigations  intended  to  discover  the  origin  of  these 
difficulties  have  not  yet  been  completed,  but  they  are  prob- 
ably due  to  some  feature  of  manipulation  of  material  during 


Art.  59.]  COPPER,    TIN,  ALUMINUM,  ZINC,  ETC.  361 

processes  of  manufacture,  including  the  treatment  of  the 
molten  metal. 

Phosphor-Bronze . 

Phosphor-bronze  possesses  merit  not  only  as  a  structural 
material  on  account  of  its  high  elastic  limit  and  ultimate 
resistance,  but  also  because  it  is  a  good  anti-friction  metal. 
Its  elastic  limit  may  be  taken  from  45,000  to  55,000  pounds 
per  square  inch  and  its  ultimate  resistance  from  50,000  to 
75,000  pounds  per  square  inch,  both  values  being  given  for 
unannealed  material.  The  same  material  as  uncmnealed 
wire  with  a  diameter  of  one-tenth  to  one-sixteenth  of  an 
inch  may  give  ultimate  resistances  varying  from  100,000 
to  150,000  pounds  per  square  inch,  or  if  annealed  not  more 
perhaps  than  50,000  to  60,000  per  square  inch.  In  the 
latter  case,  however,  the  final  stretch  may  run  from  30 
to  40  per  cent. 

Bauschinger's   Tests  of  Copper  and  Brass  as  to  Effects  of 
Repeated  Application  of  Stress. 

The  late  Professor  Bauschinger  made  some  investiga- 
tions regarding  the  effect  on  elastic  limit  and  yield  point 
of  repeated  application  of  loading  similar  to  those  made 
on  steel  and  wrought  iron.  The  grade  of  brass  used  in  his 
tests  was  called  "  red  brass." 

With  the  exception  of  one  case  of  brass  the  elastic  limit 
and  the  yield  point  were  both  materially  elevated  by 
repeated  application  of  loading,  whether  the  repetition  was 
made  without  a  period  of  rest  between  two  consecutive 
applications  or  not.  Some  repetitions  were  made  immedi- 
ately and  some  after  periods  of  i;|  to  53  hours  of  rest. 

The  effect  on  the  modulus  of  elasticity  was  small  and 
irregular,  i.e.,  in  some  cases  there  was  a  small  increase  and 


362  TENSION.  [Ch.  VII. 

in  others  a  small  decrease  and  in  some  cases  no  material 
change. 

Art.  60. — Cement,  Cement  Mortars,  etc. — Brick. 

The  ultimate  tensile  resistance  of  cements  and  cement 
mortars  depends  upon  many  conditions.  The  two  great 
divisions  of  cements,  i.e.,  natural  and  Portland,  possess  very 
different  ultimate  resistances  whether  neat  or  mixed  with 
sand,  the  latter  being  much  the  stronger.  With  given 
proportions  of  sand  or  neat,  the  ultimate  resistances  of 
cement  mortar  or  cement  will  vary  with  the  amount  of  water, 
used  in  tempering  and  with  the  pressure  under  which  the 
moulds  are  filled.  Again,  the  character  of  the  sand  used 
will  obviously  influence  largely  the  tensile  resistance  of  the 
mortar  produced,  and  not  only  the  degree  of  cleanliness, 
but  the  size  of  grain  and  the  variety  of  sizes  are  elements 
which  must  be  considered.  It  has  also  been  maintained  by 
some  that  silica-sand  will  give  better  results,  other  things 
being  equal,  than  other  sand.  Finally,  the  shape  of 
briquette  used  will  affect  the  results  to  some  extent.  Fig.  i, 
on  page  370,  shows  the  form  of  briquette  recommended  by  the 
Committee  of  the  American  Society  of  Civil  Engineers,  and 
it  is  the  form  generally  used  in  American  practice.  It  is 
foreign  to  the  purpose  of  this  work  to  enter  into  the  consider- 
ation of  all  these  influences;  they  are  only  mentioned  to 
enable  the  few  typical  experimental  results  which  follow 
to  be  interpreted  properly. 

As  the  fineness  of  grinding  is  an  important  quality  of  a 
cement,  it  is  usually  noted  by  stating  the  percentage  of 
weight  of  the  cement  which  either  passes  through  or  is 
retained  upon  a  sieve  having  a  stated  number  of  meshes 
per  linear  inch,  which  number  squared  gives  the  number 
of  meshes  per  square  inch.  The  sizes  of  the  grains  of  sand 


Art.  60.]  CEMENT,  CEMENT  MORTARS,  ETC.— BRICK.  363 

used  are  graded  in  the  same  way.  The  "  No."  of  a  sieve 
to  which  reference  may  be  made  in  what  follows  indicates, 
therefore,  the  number  of  meshes  per  linear  inch. 

Modulus  of  Elasticity. 

In  consequence  of  the  fact  that  cement,  mortars,  and 
concrete  begin  to  exhibit  permanent  stretch  at  compara- 
tively low  tensile  stresses  there  is  a  little  uncertainty  as  to 
the  value  of  the  modulus  of  elasticity  unless  distinct  state- 
ment is  made  of  the  intensities  of  stress  at  which  those 
values  are  obtained,  and  whether  the  total  stretch  is 
used  or  that  total  less  the  permanent  set.  It  is  not  possible 
to  make  such  statement  in  connection  with  all  the  values 
which  follow,  except  that  they  have  been  reached  at  low 
intensities  of  stress  unless  otherwise  stated,  and  with 
elongations  which  may  be  considered  wholly  elastic.  Al- 
though cement  mortars  and  concrete  do  not  exhibit  a  per- 
fectly elastic  behavior  their  stress-strain  lines  for  intensities 
of  stress  even  exceeding  those  used  in  practice  are  essentially 
straight  and,  on  the  whole,  exhibit  elastic  properties  at 
least  equal  to  those  of  cast  iron. 

Comparatively  few  tests  have  been  made  to  determine 
either  the  tensile  or  compressive  modulus  of  elasticity  of 
cement,  mortar  and  concrete,  although  that  quantity  is  a 
most  important  element  in  the  theory  and  design  of  much 
concrete  work  and  reinforced  concrete  members.  Mr.  W.  H. 
Henby  of  St.  Louis,  made  a  number  of  determinations  of 
the  tensile  modulus  of  elasticity  of  Portland  cement  con- 
crete of  1-2-4,  1-2-5,  i "3~6,  and  1-4-8  mixtures  and  gave 
the  results  in  a  paper  read  before  the  Engineers  Club  of 
St.  Louis  in  1900.  He  obtained  values  varying  from  less 
than  2,000,000  to  8,360,000.  Other  tests,  however,  indi- 
cate that  values  above  perhaps  3,000,000  should  not  be 


364  TENSION.  [Ch.  VII. 

used.  While  higher  values  of  the  modulus  of  elasticity  for 
rich  mixtures  of  concrete  may  exist,  the  more  important 
considerations  of  design  usually  bear 'upon  work  in  which 
concrete  must  take  serious  loading  when  less  than  thirty 
days  of  age. 

For  all  these  reasons  it  will  seldom  be  advisable  to  take 
the  modulus  of  elasticity  of  even  as  rich  a  mixture  as  i 
cement,  2  sand,  and  4  broken  stone  higher  than  about 
2,500,000,  and  it  will  be  seen  later  that  in  concrete  steel 
work  where  portions  of  a  structure  are  liable  to  be  loaded 
to  a  material  extent  within  a  comparatively  short  time 
after  removal  of  the  forms,  it  is  the  usual  practice  to  consider 
the  modulus  as  having  a  value  of  2,000,000  only.  These 
considerations  are  confirmed  by  the  results  of  tests  given 
below. 

Professor  W.  Kendrick  Hatt,  of  Purdue  University, 
in  a  paper  read  before  the  American  Section  of  the  Inter- 
national Association  for  Testing  Materials,  at  its  con- 
vention, 1902,  gave  the  following  values  for  the  tensile 
coefficient  of  elasticity  and  ultimate  tensile  resistances  of 
Portland  cement  concrete  composed  of  i  cement,  2  sand, 
and  4  broken  stone  at  the  ages  of  25,  26,  28,  and  33  days: 


Coefficient  of  Elasticity, 
Lbs.  per  Sq.  in. 

Ultimate  Tensile 
Resistance, 
Lbs.  per  Sq.  In. 

JVta.ximu.TO 

2,7OO,OOO 

'  160 

Average               .                  

2,IOO,OOO 

•JII 

Minimum    

I,4OO,OOO 

280 

It  will  be  found  in  discussing  the  compressive  modulus 
of  elasticity  that  both  moduli  probably  acquire  nearly 
their  full  value  in  about  three  months'  time.  It  would 
appear  that  moduli  do  not  increase  in  value  with  the  lapse 
of  time  to  the  same  extent  as  the  ultimate  resistance  to 


Art.  60.]          CEMENT,  CEMENT  MORTARS,  ETC.— BRICK.  365 

compression,  although  conclusive  data  as  to  this  point  are 
not  complete. 

Such  tests  as  have  been  made  show  that  the  modulus 
of  elasticity  in  tension  or  compression  for  cinder  concrete 
should  not  be  taken  higher  than  about  1,250,000  for  1-2-5 
mixtures.  Some  tests  show  somewhat  lower  values  and 
others  values  running  over  2,000,000,  but  the  latter  results 
are  too  high  for  cinder  concrete  as  ordinarily  made  and 
put  in  place. 

Ultimate  Resistance. 

The  ultimate  resistances  of  neat  Portland  cement  and 
mortar  made  with  the  same  cement  have  been  somewhat 
increased  within  the  past  half  dozen  years;  but,  upon  the 
whole,  those  resistances  as  exhibited  in  the  following 
tables  are  fairly  representative  of  the  best  grades  of  cement 
used  at  the  present  time  (1915).  The  conditions  of  manu- 
facture are  now  so  well  controlled  that  a  high  7 -day. or 
28-day  test  cement  may  readily  be  produced;  but  that 
is  not  always  desirable ;  the  main  purpose  in  masonry 
construction  being  rather  the  attainment  of  an  ultimate 
resistance  possibly  less  high  under  a  short-time  test  but 
which  continues  to  increase  indefinitely.  A  cement  show- 
ing a  high  ultimate  resistance  on  a  short-time  test  may  not 
continue  to  increase  its  ultimate  resistance  satisfactorily, 
or  that  resistance  may  even  recede  for  a  time. 

The  following  tabular  statement  is  of  interest  and  value 
as  indicating  the  character  of  the  cement  used  in  the  con- 
struction of  the  first  subway  for  the  Rapid  Transit  Railroad 
in  the  City  of  New  York.  It  will  be  observed  that  the 
ultimate  resistances  of  both  the  neat  cement  and  the  mix- 
ture of  i  cement,  2  sand,  are  practically  as  found  a  dozen 
years  later.  The  number  of  briquettes  broken  during  the 
vears  1900  and  1901  was  over  18,000.  The  average  ulti- 


366 


TENSION. 


[Ch.  VII. 


mate  tensile  resistances  in  pounds  per  square  inch  found 
by  that  series  of  tests  of  both  Portland  and  natural  cements, 
as  given  in  the  report  of  the  Chief  Engineer,  are  the 
following : 


Year. 

Neat  Cement. 

Sand  2,  Cement  i* 

i  Day. 

7  Days. 

28  Days. 

7  Days. 

28  Days. 

Portland: 
Average  result  
Average  result  
Spec,  requirements.  . 
Natural: 
Average  result  
Average  result  
Spec,  requirements  .  . 

1900 
1901 

229 
300 
150 

552 
645 
4OO 

172 
215 
125 

7H 
763 
500 

249 
322 
2OO 

276 
380 
2OO 

118 
218 

100 

434 
525 
300 

215 
350 
150 

1900 
1901 

*  For  natural  cement  a  i  cement  i  sand  mortar  was  used. 

The  results  for  the  natural  cement  are  of  interest,  as 
that  material  has  at  present  (1915)  practically  disappeared 
from  use  in  consequence  of  the  low  prices  for  which  Portland 
can  be  produced. 

Table  I  exhibits  the  results  of  tests  of  briquettes  of 
different  brands  of  domestic  Portland  cement  as  made  in 
the  testing  laboratory  of  the  Bureau  of  Surveys  of  the 
Department  of  Public  Works  of  Philadelphia,  Pa.,  for 
the  year  1912.  This  table  gives  the  fineness  of  the  cements 
in  terms  of  the  percentages  by  weight  which  were  retained 
on  sieves  with  2500,  10,000  and  40,000  meshes  per  square 
inch;  it  also  shows  the  amount  of  water  used  for  the 
different  mixtures,  as  well  as  the  specific  gravities  of  the 
material.  It  will  be  observed  that  the  briquettes  were  made 
of  neat  cement  and  of  mortar  with  a  mixture  of  i  cement 
to  3  sand.  The  results,  therefore,  show  the  effect  of  the 
presence  of  sand  on  the  ultimate  tensile  resistance  of  the 
matrix.  The  periods  at  which  the  briquettes  were  tested 
are  the  standard  24  hours,  7  days  and  28  days. 


Art.  60.]          CEMENT,  CEMENT  MORTARS,  ETC.— BRICK. 


367 


TABLE    I. 

Average  Results  of  Tests  of  Portland  Cement  Made  during  1912 — Phila.,  Pa. 


Brand 

Number  of 
Briquettes. 

Fineness  in  per 
cent. 

if 

o;O 

Percent  of 
Water  Neat 

Tensile  'strength  in  pounds 
per  square  inch. 

No. 
50. 

No. 

IOO. 

No. 

200. 

Neat 

1:3 

hrs. 

7 
dys. 

28 
dys. 

dys. 

28 

dys. 

Allentown  .  .  . 
Alpha. 

816 
500 
1  68 
388 
582 
532 
630 
28 
2,026 
1,956 
28 
42 
5H 
572 
1,984 
150 
1,162 

0.0 
0.0 
0.0 
0.0 
0.0 
O.  I 
0.2 
0.0 
0.0 
0.0 
0.0 

0-3 
0.0 
0.0 
0.0 
0.0 
0.0 

3-6 
4-8 
4-i 
3-4 

2-3 

4-0 

2.7 

4-3 
3-i 
i-9 
3-4 
5-3 
4.8 
3-0 
3-i 
3-5 
3-5 

19.7 

23-5 
23.2 
20.5 
I7.8 
20-9 
18.9 
22.8 
19.4 

16.8 

22.  I 
21.8 
23-2 
20.1 
21.  I 
21.7 
21  .2 

3-174 
3.I6I 

3-I5I 
3-I30 
3.128 
3.106 

3-II4 
3.202 
3.172 
3-I5I 
3.138 
3.082 
3-I56 
3.146 
3.127 

3.I65 
3-155 

20.0 
19.9 
19.8 
20.3 

20.  6 

21.6 

23.1 
20.  o 
20.  o 
20.  6 

20.0 

23-3 

20.0 
20.7 
19.9 
20.0 
2O.  O 

377 
399 
480 

434 
434 
398 
261 
563 
363 
453 
497 
355 
446 

372 
277 
267 
429 

721 
701 
656 
710 

767 
704 
598 
672 

752 

776 
727 
644 
686 
670 
706 
717 
713 

797 
770 

74i 
74i 
820 
718 
670 

75i 
812 
830 
812 
674 
735 
723 
80  1 
746 
759 

379 
367 
348 
384 
376 
370 
334 
366 
410 
403 
393 
328 

373 
364 
327 
376 
389 

499 
450 
430 
468 
450 
436 
412 
398 
498 

467 
445 
429 
475 
456 
436 
467 
477 

Atlas 

Bath  
Dexter  
Dragon  
Edison  
Giant 

Lehigh  

Nazareth.  .  .  . 
Northampton 
Paragon  
Penn  Allen.  .  . 
Phoenix  
Saylor's  
Vulcanite.  .  .  . 
Whitehall  .  .  . 

Table  II  shows  the  maximum,  mean  and  minimum 
results  of  the  tests  of  briquettes  of  various  brands  used  by 
the  Board  of  Water  Supply  of  the  City  of  New  York  during 
1914.  During  the  past  few  years  American  Portland  cement 
has  been  improved  in  uniformity  of  quality  and  fineness  of 
grinding.  These  tests,  therefore,  show  the  latest  results 
of  the  best  practice  in  cement  production  and  use.  The 
tabulated  values  show  the  variations  occurring  in  systematic 
testing  of  large  quantities  of  cements  at  7 -day  and  28-day 
periods.  The  results  are  all  in  pounds  per  square  inch  and 
so  arranged,  as  is  evident,  that  in  each  vertical  group  of 
three  in  each  column  the  highest  value  is  the  maximum  and 
the  lowest,  the  minimum,  the  mean  occupying  the  middle 
position. 


368 


TENSION. 
TABLE   II. 


[Ch.  VII. 


Brand. 

Approx. 
Bbls. 

No. 
Briquettes. 

Neat 
Lbs.  per  Sq.  In. 

i  c.  3  Ottawa  Sand 
Lbs.  per  Sq.  In. 

7  Day. 

28  Day. 

7  Day. 

28  Day. 

Alpha 

I70,OOO 
324,000 
520,OOO 

45,000 

'•- 
852   • 

I62O 
2790 
243 

f 

866 
700 
572 
744 
615 
453 
755 
643- 
549 
815 
7H 
669 

857 
716 
601 
864 

747 
650 

785 
662 

575 
883 

773 
694 

320 
285 
246 
263 
192 
147 
324 
279 
221 

265 
247 
208 

458 
380 
336 
360 
300 
2I9 
414 
356 
279 
392 

354 
322 

Alsen 

Atlas  

Saylor's.  ...  r  .... 

The  large  quantities  of  cement  used  with  the  corre- 
sponding large  number  of  briquettes  tested  give  the  Table 
special  value  and  interest.  The  Ottawa  sand  is  the 
standard  silica  sand  of  that  name  so  extensively  used  in 
cement  mortar  testing. 

The  preceding  tabular "  results  give  ultimate  tensile  re- 
sistances for  periods  no  longer  than  28  days,  but  both  neat 
cement  and  cement  mortars  go  on  acquiring  additional 
resistance  for  long  periods,  although  at  slow  rates  after  a 
period  of  28  days;  indeed,  it  may  be  stated  without  ex- 
aggeration generally  after  a  period  of  only  7  days.  Table 
III  therefore  is  used  to  show  the  increase  of  ultimate  resist- 
ance up  to  a  period  of  six  months.  The  results  of  this  table 
are  taken  from  the  Annual  Report  of  the  Bureau  of  Surveys 
of  Philadelphia,  Pa.,  for  the  year  1901.  It  will  be  observed 
tliat  the  values  are  not  greatly  different  from  those  given 
in  Table  I  at  a  date  10  years  later.  In  fact,  the  earlier 
values  are  a  little  higher  than  the  later,  showing  the  ten- 
dency to  secure  a  higher  degree  of  permanency  in  the  setting 
of  the  cement  rather  than  higher  ultimate  tensile  resistances. 


Art.  60.]         CEMENT,  CEMENT  MORTARS,  ETC.— BRICK. 


369 


TABLE    III. 

AVERAGE  RESULTS   OF   PORTLAND   CEMENT   TESTS  MADE 
DURING  1901. 


Brand. 

No.  of   ' 
Tests. 

Ultimate  Tensile  Resistance  in  Pounds  per  Square  Inch. 

Neat. 

Broken. 

24  Hrs. 

7  Days. 

28  Dys. 

2  Mos. 

3  Mos. 

4  Mos. 

6  Mos. 

Alpha  

136 

820    . 
16 

28 
816 

72 

112 

36 

16 
764 
4,012 
204 
356 

357 
542 
235 
363 
424 
418 
377 
345 
460 

295 
437 
290 

524 

770 
728 
336 
826 
669 
830 
699 
721 
800 
697 
721 
748 
713 

834 
790 
387 
932 
719 
864 
747 
723 
955 
766 
746 
767' 
765 

885 
802 

443 
778 
713 
775 
684 

775 
756 
73i 
707 

788 

813 
76l 

745 
735 

766 

7i5 
807 
796 

785 
815 

776 

774 

733 
727 
710 

775 

827 
825 

786 
760 

745 
740 

Atlas   

*  Castle 

Dexter 

Giant 

Kra  use's  
Lehigh 

Phoenix 

Reading  

Savior's 

Star  

Vulcanite  
Whitehall  

Brand. 

No.  of 
Tests. 

Ultimate  Tensile  Resistance  in  Pounds  per  Square  Inch. 

i  to  3  Standard  Quartz  Sand. 

Broken. 

24  Hrs. 

7  Days. 

28  Dys. 

2  Mos. 

3  Mos. 

4  Mos. 

6  Mos. 

Alpha  
Atlas           

I36 
820 

16 

28 
816 

72 

112 
36 

16 

764 
4,012 
204 
356 

8l 
104 

65 
68 

87 
74 
76 

94 
150 
64 

77 
45 
87 

252 
204 

121 

298 
227 
229 
233 
264 
263 
217 
219 
226 
232 

3H 
289 
176 
336 
309 
285 
329 
343 
301 
296 
298 
287 
313 

344 
324 
215 
312 
328 
270 
296 

338 
319 
321 
269 
295 

312 
321 

317 
310 

301 
301 
298 
295 

302 

337 

328 
303 

3ii 
286 
280 
343 

262 
308 

329 

325 

286 
330 

*  Castle  
Dexter  
Giant  
Krause's.  ...  .  .. 
Ivehigh              .  . 

Phoenix            .  . 

Reading  
Say  lor'  s  
Star  
Vulcanite  
Whitehall  

370 


TENSION. 


[Ch.  VII. 


During  the  construction  of  a  number  of  dams  in  the 
Croton  basin  supplying  the  water  works  of  the  City  of  New 
York,  briquettes  of  neat  cement  and  of  mortar  i  to  2  and 
i  to  3  were  tested  after  periods  beginning  with  one  week 
and  extending  up  to  five  years.  There  was  a  continuous 
increase  of  ultimate  resistance  throughout  the  entire  period, 
although  at  a  very  slow  rate  after  about  six  months.  At 
the  end  of  five  years  the  neat  Portland  cement  attained 
an  ultimate  resistance  of  840  pounds  per  square  inch  and 
the  i  to  2  mortar,  700  pounds  per  square  inch,  while  the 
i  to  3  mortar  reached  590  pounds  per  square  inch. 

Other  tests  of  briquettes  up  to  two  years  of  age  and 
more  confirm  the  preceding  results. 

The  recent  cement  product,  called  silica-Portland 
cement,  is  manufactured  by  grinding  together  certain 
portions  of  clean  silicious  sand  and  Portland  cement. 
The  results  given  below  are  taken  from  the  tests  of  such 
silica-Portland  cement,  manufactured  by  the  Silica- Port- 
land Cement  Co.,  of  Long  Island  City,  N.  Y.  One  part, 
by  weight,  of  Aalborg  Portland  cement  was  ground  to- 

'  TABLE  IV.. 

SILICA-PORTLAND  CEMENT. 

Ultimate  Tensile  Resistance  in  Pounds  per  Square  Inch. 


Ag 

e. 

Mixture. 

Per  Cent, 
of  Water. 

Seven 
Days. 

Fifteen 
Days. 

Twenty-one 
Days. 

Two 
Hundred  and 
Nineteen 
Days. 

Neat  

18-21% 

{148 
I30 

(172 
6-j  165 

(  166 
8  •<  149 

121 

1   121 

(1-6)  s.  c.-2  q.  .. 

"% 

(     8l 
23-^     69 

I     58 



s\  "s 

(    88 

522O 
204 
194 

All  specimens  one  day  in  air  and  remainder  in  water. 


Art.  60.]          CEMENT,  CEMENT  MORTARS,  ETC.—  BRICK.  371 

gether  with  six  parts,  by  weight,  of  clean  silicious  sand 
to  such  a  degree  of  fineness  that  essentially  all  of  the 
product  passed  through  a  32,ooo-mesh  sieve.  This  finely 
ground  mixture  of  i  cement  to  6  sand,  by  weight,  is  called 
"neat  "  in  what  follows,  while  "(1-6)5.  c.-2  q."  is  i  part,  by 
weight,  of  the  "neat"  silica-Portland  cement  to  2  parts,  by 
weight,  of  crushed  quartz,  or  "standard"  sand,  all  of  which 
passes  a  No.  20  sieve  and  is  retained  on  a  No.  30  sieve.  The 
results  were  obtained  in  the  cement-testing  laboratory  of 
the  department  of  civil  engineering  of  Columbia  University. 
The  figures  on  the  left  of  the  brackets  show  the  number  of 
tests  of  which  the  ultimate  resistances  are  the  greatest, 
mean,  and  least  in  each  case. 

Five  seven-day  tests  of  the  Aalborg  Portland  cement 
used  in  the  manufacture  of  the  silica-Portland  cement 
gave  the  following  greatest,  mean,  and  least  ultimate 
tensile  resistances,  the  specimens  having  been  one  day  in 
air  and  six  days  in  water: 

Greatest.  Mean.  Least. 

594  Ibs.  per  sq.  in.  536  Ibs.  per  sq.  in.  441  Ibs.  per  sq.  in. 

Four  specimens  of  the  neat  silica-Portland  cement  (1-6), 
one  day  in  air  and  the  remainder  of  the  time  in  water, 
gave  the  following  results: 

Age. 
308  Ibs.  per  sq.  in  ......  199  days. 

Neat  (1-6)..     .  J  264   "    "   .....  '9°  " 
|  294  .....  189 

"    "   .....  185  " 


All  the  preceding  tensile  tests  of  cement  and  cement 
mortars,  unless  otherwise  stated,  were  made  with  the  shape 
of  briquette  shown  in  Fig.  i,  which  was  recommended  for 
use  in  the  report  of  the  "  Committee  on  a  Uniform  System 
for  Tests  of  Cement  "  of  the  American  Society  of  Civil 
Engineers.  That  report  was  made  in  1912,  and  the  bri- 


372 


TENSION. 


[Ch.  VII. 


quette  recommended  has  become  the  standard  in  American 
practice  for  the  testing  of  cements  and  mortars. 


FIG.  i. 


Weight  of  Concrete. 

As  concrete  is  frequently  used  in  masses  where  weight 
is  an  important  element,  it  is  always  desirable  to  use  an 
aggregate  of  high  specific  gravity.  Concrete  when  made  of 
cement,  sand  and  silicious  gravel  or  broken  limestone,  trap- 
rock  or  granitic  rock  in  such  mixtures  as  are  commonly 
employed,  will  weigh  from  140  to  155  pounds  per  cubic  foot 
with  the  greater  part  running  from  145  to  150  pounds  per 
cubic  foot. 

The  weight  of  cinder  concrete  will  necessarily  vary  much 
with  the  character  of  the  cinders.  It  may  usually  be  taken 
as  weighing  about  two-thirds  as  much  as  ordinary  concrete 


Art.  60.  CEMENT,  CEMENT  MORTARS,  ETC.— BRICK.  373 

made  with  gravel  or  broken  stone,  i.e.,  from  100  to  no 
pounds  per  cubic  foot. 

Adhesion  between  Bricks  and  Cement  Mortar. 

General  Q.  A.  Gillmore  many  years  ago  investigated 
the  adhesion  of  bricks  to  the  cement  mortar  joint  between 
them  and  also  the  adhesion  of  fine-cut  granite  to  a  similar 
joint.  As  might  be  expected  in  connection  with  such  tests 
his  results  varied  greatly,  the  highest  belonging  to  a  rich 
cement  mortar  and  the  lowest  to  the  lean  mortar  of  i 
cement  to  6  sand.  He  found  the  adhesion  to  vary  from 
about  31  pounds  per  square  inch  for  neat  cement  to  brick 
to  nearly  3.3  pounds  per  square  inch  for  a  lean  mortar  of 
i  cement  to  6  sand.  With  fine-cut  granite  the  adhesion 
for  neat  cement  was  27.5  pounds  per  square  inch  and  for 
cement  mortar  of  i  cement  to  4  sand  about  8  pounds  per 
square  inch.  It  is  highly  probable  that  the  actual  adhesion 
of  bricks  and  cut  stone  to  the  usual  joints  made  of  i  cement 
to  2  sand  or  i  cement  to  3  sand  would  be  materially  less 
in  a  mass  of  masonry  than  as  arranged  for  a  laboratory 
test.  Nevertheless  these  early  investigations  would  indi- 
cate that  such  joints  might  be  worth  from  8  to  12  pounds 
per  square  inch  for  bricks  and  but  little  different  for 
granite. 

Mr.  Emil  Kuichling  prepared  a  paper  in  1888  from  all 
available  sources  for  the  purpose  of  disclosing  what  all 
experimental  investigation  had  determined  up  to  that  time. 
These  results  indicated  that  neat  cement  might  give  ad- 
hesion to  bricks  or  cut  stone  varying  from  about  20  pounds 
up  to  over  200  pounds  per  square  inch,  with  values  from 
29  pounds  up  to  146  pounds  per  square  inch  for  mortar  of 
i  cement  to  i  sand;  and  38  pounds  to  73  pounds  per  square 
inch  for  a  mortar  of  i  cement  to  2  sand.  Further,  accord- 
ing to  his  table  a  mortar  of  i  cement  to  3  sand  would 


374 


TENSION. 


[Ch.  VII. 


yield  adhesion  from  13  pounds  up  to  48  pounds  per  square 
inch  and  but  little  less  for  a  mortar  of  i  cement  to  4  sand. 
Nearly  all  these  results,  however,  are  undoubtedly  too  high 
for  the  usual  masses  of  masonry  in  engineering  construction. 

Other  experimental  determinations  of  the  adhesive 
resistance  of  natural  and  Portland  cement  mortars  to 
brick  and  stone  may  be  found  in  the  report  of  the  Chief  of 
Engineers,  U.  S.  A.,  for  1895.  At  the  age  of  28  days 
the  adhesive  resistance  of  neat  Portland  cement  to  the 
surface  of  sawn  limestone  was  about  270  pounds  per  square 
inch;  about  240  pounds  per  square  inch  with  a  mortar  of 
i  cement  to  J  sand;  about  225  pounds  per  square  inch 
with  a  mortar  of  i  cement  to  i  sand,  and  about  170  pounds 
per  square  inch  with  a  mortar  of  i  cement  to  2  sand. 

Table  V  exhibits  the  average  results  of  three  and  six 
months'  tests  of  the  adhesion  of  Portland  and  natural 
cement  mortars  to  bricks  which  were  cemented  to  each 
other  at  right  angles  and  then  pulled  apart  normally  at  the 
ends  of  the  periods  named.  These  average  results  are 
taken  from  the  same  report  of  the  Chief  of  Engineers, 
U.  S.  A,,  for  1895. 

TABLE  V. 

AVERAGE  ADHESIVE  RESISTANCE  OF  BRICKS  CEMENTED 
TOGETHER  AT  RIGHT  ANGLES  TO  EACH  OTHER. 


Cement. 

Mortar. 

Adhesion, 
Pounds  per  Square  Inch. 

Portland 

Neat 

60 

c. 

is. 

60 

c. 

I  S. 

40 

c. 

2  S. 

20 

Na 

ural 

c. 

Nc 

38- 

at 

20 

55 

c. 

is. 

50 

c. 

I  S. 

45 

i  • 

(C 

c. 
c. 

2  S. 

3S. 

30 
15 

Art.  60.]          CEMENT,  CEMENT  MORTARS,  ETC.— BRICK.  375 

There  will  also  be  found  in  that  report  average  values 
of  the  shearing  adhesion  of  plain  i-inch  round  bolts  to  neat 
Portland  cement  and  to  Portland  cement  mortars  of  i 
month's  age,  the  bolts  having  been  embedded  at  various 
depths  from  2  to  10  inches  in  the  mortars.  The  shearing 
adhesion  for  the  neat  cement  varied  from  a  maximum 
of  345  pounds  per  square  inch  for  a  depth  of  insertion  of  4 
inches  down  to  230  pounds  per  square  inch  for  a  depth  of 
insertion  of  about  8J  inches.  In  the  case  of  the  Portland 
cement  mortar  of  i  cement  to  2  sand  the  shearing  adhesion 
varied  from  a  maximum  of  280  pounds  per  square  inch  for  a 
depth  of  insertion  of  the  bolt  of  2^  inches  down  to  250 
pounds  per  square  inch  for  a  depth  of  insertion  of  about 
7 f  inches.  When  the  bolt  was  embedded  in  the  Portland 
cement  mortar  of  i  cement  to  4  sand  the  shearing  adhesion 
ranged  from  a  maximum  of  about  145  pounds  per  square 
inch  for  a  depth  of  insertion  of  10  inches  to  a  minimum  of 
about  70  pounds  per  square  inch  for  a  depth  of  insertion 
of  2  inches.  These  values  of  shearing  adhesion  are  impor- 
tant results  in  the  theory  and  design  of  concrete-steel 
members. 

The  Effect  of  Freezing  Cements  and  Cement  Mortars. 
There  have  been  many  attempts  made  to  determine  the 
effect  of  freezing  neat  cements  and  cement  mortars  after 
having  been  mixed  for  use  at  various  ages  and  under 
various  conditions.  vSome  valuable  data  have  been  ac- 
cumulated, but  the  conditions  attending  such  investiga- 
tions are  so  complicated  and  so  difficult  to  be  analyzed 
quantitatively  that  many  most  discordant  conclusions  have 
been  reached.  Different  results  will  follow  if  the  freezing 
is  done  immediately  after  the  mixing  of  the  cement  or 
mortar,  or  after  the  initial  set  has  taken  place,  or  after  the 
considerable  hardening  which  takes  place  at  the  age  of 


376  TENSION.  [Ch.  VII. 

12  to  24  hours.  Probably  the  best  data  in  this  connection 
arise  from  an  engineer's  practical  experience  in  laying 
masonry  when  the  temperature  of  the  air  is  below  the 
freezing-point.  Under  such  circumstances  it  is  rarely 
the  case  that  anything  more  than  surface  freezing  takes 
place  before  the  hardening  of  Portland  cement.  With  the 
slower  action  of  the  natural  cements  similar  conditions  do 
not  exist.  It  is  undoubtedly  prejudicial  even  with  Port- 
land cements  to  have  alternate  freezing  and  thawing  take 
place  at  comparatively  short  intervals  of  time.  On  the 
other  hand,  the  great  majority  of  laboratory  investigations 
indicate  that  Portland  cement  or  cement  mortars  may  be 
severely  frozen  and  remain  so  for  long  periods  of  time 
without  essential  injury.  It  is  probable  that  setting  usually 
proceeds  during  a  frozen  condition,  but  at  an  exceedingly 
slow  rate,  and  that  the  operation  of  setting  is  actively 
renewed  after  thawing. 

While  it  has  been  stated  in  some  quarters  that  natural 
cements  may  be  frozen  similarly  and  thawed  without 
essential  injury,  there  is  considerable  laboratory  evidence 
as  well  as  that  of  practice  which  indicates  that  conclusion 
to  be  erroneous,  especially  if  it  be  given  any  considerable 
application.  There  may  be  cases  in  which  natural  cements 
can  be  or  have  been  frozen  without  essential  injury,  but 
the  author's  experience  in  extended  practical  operations  in 
masonry  construction  induces  him  to  believe  that  any 
natural  cement  severely  frozen  before  being  thoroughly 
hardened  is  so  seriously  injured  as  to  be  practically  de- 
stroyed. On  the  other  hand,  his  extended  observations 
not  only  on  his  own  work,  but  on  those  of  others,  lead  him 
to  believe  that,  as  a  rule,  Portland  cement  will  not  be 
sensibly  injured  under  the  conditions  of  actual  masonry 
construction  by  being  frozen.  It  is  customary  in  most 
large  works  to  permit  no  masonry  to  be  laid  at  a  tempera- 


Art.  60.]          CEMENT,   CEMENT  MORTARS,   ETC.— BRICK.  377 

ture  much  below  about  26°  Fahr.  above  zero,  but  with 
precautions  easily  attained  it  is  certain  that  concrete  and 
other  masonry  laid  in  Portland  cement  mortar  may  prop- 
erly and  safely  be  put  in  place  several  degrees  below  that 
temperature. 

It  has  also  been  stated  in  some .  quarters  that  natural 
cements  and  some  Portlands  have  been  actually  improved 
by  being  frozen.  Such  conclusions  should  be  received  with 
exceeding  caution.  The  author  believes  that  there  is  no 
conclusive  evidence  that  any  cement  or  cement  mortar  can 
be  improved  by  freezing. 

In  cold  weather  it  is  customary  on  some  works  to  use 
salt  water  for  mixing  mortars  and  concretes,  and  that 
practice  when  suitably  conducted  may  be  resorted  to  with 
safety  and  propriety.  Such  solutions  generally  run  from 
2  to  8  or  10  per  cent,  by  weight  of  salt.  Occasionally,  also, 
soda  is  dissolved  in  water  at  the  rate  of  2  pounds  per  gal- 
lon. Before  using  this  solution  an  equal  volume  of  water 
is  added  so  that  the  final  solution  contains  about  i  pound 
of  soda  to  a  gallon  of  water.  This  solution  expedites 
the  setting  of  the  cement  with  a  view  to  accomplishing 
a  safe  degree  of  hardening  before  the  mortar  is  frozen. 
It  is  doubtful  whether  this  practice  should  be  encouraged. 

Tke  Linear  Thermal  Expansion  and  Contraction  of 
Concrete  and  Stone. 

Satisfactory  investigations  regarding  the  expansion  and 
contraction  of  concrete  and  stone  are  exceedingly  few  in 
number,  and  the  data  by  which  variations  in  the  dimen- 
sions of  large  masses  of  masonry  due  to  temperature  changes 
can  be  computed  are  correspondingly  meagre.  Professor 
William  D.  Pence,  of  Purdue  University,  has  made  such 
investigations  and  presented  the  results  in  a  valuable 
paper  read  before  the  Western  Society  of  Engineers, 


37* 


TENSION. 


(Ch.  VII. 


November  20,  1901.  In  his  experimental  work  he  com- 
pared the  thermal  linear  changes  of  concrete  bars  and  bars 
of  steel  and  copper,  basing  the  coefficients  of  expansion  of 
the  concrete  and  mortar  on  the  relative  changes  of  the 
two  materials  for  the  same  range  of  temperature.  These 
experiments  were  conducted  with  great  care,  but  the 
resulting  values  might  perhaps  have  been  at  least  better 
denned  had  two  materials  been  employed  with  a  greater 
difference  in  their  rates  of  thermal  expansion  and  contrac- 
tion. Professor  Pence  employed  two  kinds  of  concrete  and 
one  bar  of  Kankakee  limestone,  seven  experiments  having 
been  performed  on  a  concrete  of  i  Portland  cement,  2  sand, 
and  4  broken  stone ;  one  on  a  concrete  of  i  Portland  cement, 
2  sand,  and  4  gravel ;  and  three  on  a  concrete  composed  of  i 
cement  and  5  of  sand  and  gravel,  making  the  mixture 
essentially  equivalent  to  the  preceding  concrete  of  i  cement, 
2  sand,  and  4  gravel.  The  maximum,  mean,  and  minimum 
coefficients  of  linear  expansion  per  degree  Fahr.  found  in 
these  tests  were  as  follows: 


Kind  of  Concrete. 

Maximum. 

Mean. 

Minimum. 

Broken  stone     1:2:4  

OOOOOS7 

0000055 

0000052 

Gravel    1*2*4. 

OOOOO^A 

Gravel    i  '  5                 

OOOOOS5 

OOOOO^T. 

0000052 

Kankakee  limestone 

OOOOO56 

Between  January  and  June,  1902,  Messrs.  J.  G.  Rae 
and  R.  E.  Dougherty,  graduating  students  in  Civil  Engineer- 
ing at  Columbia  University,  with  the  aid  of  Professor 
Hallock  of  the  Department  of  Physics  of  the  same  university, 
determined  with  great  care  by  the  most  accurate  direct 
measurements  the  coefficients  of  linear  thermal  expansion 
of  one  bar  of  concrete  of  i  Portland  cement,  3  sand  and  5 
gravel,  and  one  bar  of  mortar  of  i  Portland  cement  and  2 
sand,  each  bar  being  4  inches  by  4  inches  in  cross-section 


Art.  61.]  TIMBER  IN    TENSION.  379 

and  about  3  feet  long,  both  bars  being  tested  at  the  age 
of  about  5^  years.  The  coefficients  of  linear  thermal 
expansion  for  each  degree  Fahr.  found  in  these  investiga- 
tions were  as  follows: 

For  1:3:5  concrete 00000655 

"     1:2  mortar 0000056 1 

It  is  believed  that  these  last  two  determinations  were 
made  with  the  utmost  accuracy  attainable  at  the  present 
time  in  an  unusually  well  equipped  physical  laboratory 
and  under  most  favorable  conditions. 

When  it  is  remembered  that  the  coefficient  of  linear 
thermal  expansion  of  such  iron  and  steel  as  are  used  in 
engineering  structures  is  about  .0000066,*  it  is  apparent 
that  structures  of  combined  concrete  or  other  masonry  and 
steel  may  be  expected  to  act  under  thermal  changes  essen- 
tially as  a  unit,  a  conclusion  which  is  justified  at  the  present 
time  by  extended  experience. 

Art.  61.— Timber  in  Tension. 

The  ultimate  resistance  of  timber  in  general  is  much 
affected  by  the  moisture  which  it  contains,  except  that  the 
amount  of  moisture  does  not  appear  to  affect  sensibly  the 
ultimate  tensile  resistance.  At  this  point,  therefore,  no 
further  attention  will  be  given  to  the  effect  of  moisture  or  sap 
on  the  tensile  resistance,  but  the  influence  of  moisture  on  the 

*  A  large  number  of  determinations  of  the  thermal  expansion  of  iron  and 
steel  per  degree  Fahr.  may  be  found  in  the  U.  S.  Report  of  Tests  of  Metals 
and  Other  Materials  for  1887.  The  maximum,  mean,  and  minimum  for 
steel  bars  are  as  follows: 

.000006756  .000006466  .00000617 

Other  coefficients  of  thermal  expansion  are  also  given  as  follows: 

Wrought  iron 00000673 

Cast  iron 000005926 

Copper t 000009129 


TENSION. 


[Ch.  VII. 


compressive  and  bending  resistances  will  be  fully  set  forth 
in  the  articles  devoted  to  timber  in  compression  and  bending. 

There  are  few  results  of  investigations  which  give  satis- 
factory moduli  of  elasticity  for  timber  in  tension.  Values 
are  given  in  the  annual  "  U.  S.  Report  of  Tests  of  Metals 
and  Other  Materials,"  but  these  results  are  generally  for 
small  selected  sticks  which  are  quite  different  from  com- 
mercial sizes  of  lumber  as  generally  used.  Some  of  these 
moduli  run  up  to  nearly  3,000,000,  which  is  much  too  high  for 
any  ordinary  commercial  timber  as  used  in  structural  work. 

In  "  Tests  of  Structural  Timbers,"  by  McGarvey  Cline, 
Director  of  Forest  Products  Laboratory,  and  A.  L.  Heim, 
Engineer  of  Forest  Products,  issued  as  Bulletin  108  of  the 
U.  S.  Department  of  Agriculture,  1912,  a  large  number  of 
determinations  are  made  of  ultimate  resistance,  elastic  limit 
and  modulus  of  elasticity  for  commercial  sizes  of  lumber 
of  nine  different  kinds  of  generally  used  timber.  The 
moduli  of  elasticity,  however,  are  determined  from  bending 
tests,  which  makes  them  a  kind  of  composite  of  both  tension 
and  compression  values.  The  results  found,  however,  are 
among  the  best  available. 

The  following  tabular  statement  gives  the  moduli  for 
green  and  air-seasoned  structural  sizes: 

TABLE    I. 


Green 

Air-seasoned 

Wt.  per 
cu.  ft. 
oven-dry 

Long-leaf  Pine 

I  46^,OOO 

705  OOO 

-2C 

Douglas  Fir 

,  SI  7.OOO 

549  ooo 

28 

Short-leaf   Pine 

,47^,000 

726  ooo 

-JQ 

Western  Larch 

,3OI,OOO 

487  ooo 

28 

Loblolly  Pine  

,387,000 

,487,000 

31 

Tamarack  

,220,000 

1,341,000 

30 

Western  Hemlock  

,445,000 

i,737,ooo 

27 

Red  Wood  

,042,000 

890,000 

-      22 

Norway  Pine  

,133,000 

1,418,000 

25 

Art.  61.]  TIMBER  IN  TENSION.  381 

It  will  be  noticed  that  redwood  gives  the  lowest 
modulus  of  elasticity  and  Norway  pine  next  above  it 
except  the  value  for  air-seasoned  tamarack.  Long-leaf 
pine,  short-leaf  pine,  and  Douglas  fir  give  nearly  the 
same  results. 

In  determining  the  tensile  resistance,  and,  indeed,  other 
resistances  of  timber,  the  size  of  the  specimen  plays  a 
more  important  part,  probably,  than  in  any  other  class 
of  materials  used  by  the  engineer.  Small  specimens,  such 
as  are  usually  employed  in  tensile  tests,  are  inevitably  so 
selected  as  to  eliminate  such  defects  as  decay  and  decayed 
or  other  knots,  wind  shakes,  season  cracks,  and  other 
deteriorating  features,  so  that  the  results  exhibit  physical 
properties  belonging  to  the  best  parts  of  full-size  sticks. 
In  engineering  practice,  on  the  other  hand,  large  pieces  of 
timber  must  be  used  as  furnished  in  the  timber  market. 
However  close  the  inspection  may  be  such  pieces  in- 
variably include  within  their  volumes  many  spots  of  weak- 
ness due  to  those  features  which  in  the  small  specimen  are 
carefully  excluded.  It  is  of  the  utmost  consequence, 
therefore,  in  dealing  with  physical  data  belonging  to  timber 
to  realize  that  results  determined  by  the  testing  of  small 
specimens  are  almost  without  exception  materially  mis- 
leading in  consequence  of  reaching  higher  values  than  those 
which  can  possibly  belong  to  the  average  stick  used  in 
structural  work.  These  observations  must  be  carefully 
remembered  in  considering  the  experimental  data  which 
follow. 

While  there  exists  a  large  amount  of  data  on  the  tensile 
tests  of  timber  it  relates  largely  to  small  selected  sticks 
or  is  otherwise  scarcely  available  for  engineering  construc- 
tion. The  best  recent  data  are  given  by  Messrs.  Cline  and 
Heim  from  which  Table  I  was  taken.  On  page  57  of  that 
Bulletin  tabulated  data  of  a  large  number  of  bending  tests 


382 


TENSION. 


[Ch.  VII. 


of  green  and  dry  structural  timbers  are  found,  the  failures 
being  by  tension  in  the  fibres  subjected  to  that  kind  of 
stress.  Those  data  are  shown  in  Table  II.  The  modulus 
of  rupture  is  simply  the  intensity  of  stress  in  the  most 
remote  fibre  of  the  timber. 


TABLE   II. 


Species. 

Average 
modulus 
of  rupture 
Ibs.  per 
sq.  in. 

Modulus  of  rupture  in  per  cent,  of  average  green 
modulus  of  rupture.     First  failure  by  tension  (per  cent.) 

All 
tension 
failures. 

Failure  due  to 

Large 
knots. 

Small 
knots. 

Irregu- 
lar grs. 

Pitch 
pockets 

Nothing 
apparent 

Long-leaf  pine: 
Green    . 

6,140 

5,749 

5,983 
6,372 

5,548 
6,573 

4,948 
5,856 

5,084 
6,118 

4,556 
5,498 

5,296 
6,420 

4,472 
3,891 

3,864 
6,054 

112 
121 

83 
82 

94 
117 

73 
no 

86 

120 

90 
1  06 

74 
1  08 

81 
80 

94 
134 

112 
121 

104 
136 

109 
132 

100 
I24 

98 
138 

98 
IOO 

81 
119 

90 

87 

105 
129 

Dry  

Douglas  fir: 
Green  
Dry  
Short-leaf  pine: 
Green  
Dry  
Western  larch  : 
Green  
Dry  
Loblolly  pine  : 
Green  
,    Dry  

82 
69 

76 
96 

1  66 
1  02 

73 
39 

80 

78 

90 

77 
82 

100 

103 

90 
114 

96 

112 

106 

55 
48 

73 

90 

85 

77 

112 

86 
90 

Tamarack  : 
Green  

Dry 

Western  hemlock: 
Green  
Dry 

77 

92 

"58" 

Redwood  : 
Green  
Dry  
Norway  pine: 
Green  
Dry 

94 

136 

The  moduli  of  rupture  are  the  averages  of  all  failures 
whether  by  tension,  compression  or  shear,  but  the  figures 
given  in  the  table  after  the  second  column  represent  the 


Art.  61.] 


TIMBER  IN  TENSION. 


383 


percentages  of  the  average  "  green  "  moduli  of  rupture  at 
which  the  extreme  fibres  failed  in  tension  uuder  influence 
of  "  large  knots,"  "  small  knots,"  "  irregular  grain  "  or 
"  nothing  apparent  "  as  indicated  at  the  head  of  each 
column.  Although  these  values  are  not  found  by  direct 
tests  of  tension,  they  may  be  accepted  as  fair  and  suitable 
ultimate  resistances  of  the  different  kinds  of  timber  in 
tension. 

TABLE  III. 


Kind  of  Timber. 

Ultimate  Resistance, 
Pounds  per 
Square  Inch. 

Working  Stresses, 
Pounds  per 
Square  Inch. 

With 
Grain. 

Across 
Grain. 

With 
Grain. 

Across 
Grain. 

White  oak 

10,000 

7,000 

12,000 
12,000 
10,000 

9,000 
9,000 
8,000 
10,000 
10,000 
8,000 
6,000 
6,000 
8,000 
9,000 
7,000 

2,000 
500 

6OO 

500 
500 

500 

I,OOO 

700 

I,2OO 
1,200 
1,000 
900 
900 
800 
1,000 
1,000 

800 

600 

600 
800 
900 
700 

200 
50 

60 

50 
50 

50 

\Vhite  pine 

Southern  long-leaf  or  Georgia  yellow 
pine 

Douglas,  Oregon,  and  yellow  fir  

\Vashington  fir  or  pine  (red  fir) 

Northern  or  short-leaf  yellow  pine.  .  .  . 
Red  pine 

Norway  pine  

Canadian  (Ottowa)  white  pine  

Canadian  (Ontario)  red  pine 

Spruce  and  Eastern  fir 

Hemlock  

Cypress  . 

Cedar      

Chestnut                   

California  redwood  

California  spruce   

Reviewing  all  the  experimental  work  which  has  been 
done  up  to  the  present  time  (1902)  in  determining  the 
ultimate  tensile  resistance  of  timber,  and  keeping  in  view 
experience  with  the  resistance  of  full-size  timber  sticks 
in  completed  structures,  the  best  representative  series  of 
values  of  the  ultimate  and  working  tensile  intensities  of 
timbers  is  that  recommended  by  the  Committee  on 


384  TENSION.  [Ch.  VII. 

"  Strength  of  Bridge  and  Trestle  Timbers  "  of  the  Associa- 
tion of  Railway  Superintendents  of  Bridges  and  Buildings 
at  the  Fifth  Annual  Convention  in  New  Orleans,  1895. 
That  series  is  given  in  Table  III. 

The  ultimate  resistances  of  the  table  are  much  too 
high  for  full  size  pieces,  but  the  working  stresses  may  be 
accepted  as  they  stand. 

It  will  be  noticed  that  the  ultimate  tensile  resistance 
of  the  various  timbers  across  the  grain,  so  far  as  they  are 
given,  are  but  small  fractions  of  the  ultimate  resistances 
along  the  grain.  A  corresponding  large  decrease  in  resist- 
ance across  the  grain  will  also  be  found  in  connection  with 
the  compressive  resistance  of  the  same  timbers.  The 
working  resistances  given  in  this  table  are  those  employed 
in  the  great  bulk  of  engineering  timber  structures. 


CHAPTER   VIII. 

COMPRESSION. 

Art.  62. — Preliminary. 

WITH  the  exception  of  material  in  the  shape  of  long 
columns,  but  few  experiments,  comparatively  speaking, 
have  been  made  upon  the  compressive  resistance  of  con- 
structive materials. 

Pieces  of  material  subjected  to  compression  are  divided 
into  two  general  classes — "  short  blocks  "  and  "long  col- 
umns ' ' ;  the  first  of  these,  only,  afford  phenomena  of  pure 
compression. 

A  "  short  block  "  is  such  a  piece  of  material  that  if  it  be 
subjected  to  compressive  load  it  will  fail  by  pure  compres- 
sion. 

On  the  other  hand,  a  long  column  (as  has  been  indi- 
cated in  Art.  35)  fails  by  combined  compression  and  bending. 

Short .  blocks  only  will  be  considered  in  the  articles 
immediately  succeeding,  while  long  columns  will  be  sepa- 
rately considered  further  on. 

The  length  of  a  short  block  is  usually  about  three  times 
its  least  lateral  dimension  or  less. 

It  has  already  been  shown  in  Art.  5  that  the  greatest 
shear  in  a  short  block  subjected  to  compression  will  be 
found  in  planes  making  an  angle  of  45°  with  the  surfaces 
of  the  block  on  which  the  compressive  force  acts,  i.e.,  with 

385 


386  COMPRESSION.  [Ch.  VIII 

its  ends.  If  the  material  is  not  ductile  this  shear  will 
frequently  cause  wedge-shaped  portions  to  separate  from 
the  block.  But  the  friction  at  these  end  surfaces,  and  in 
the  surfaces  of  failure  will  prevent  those  wedge  portions 
shearing  off  at  that  angle.  In  fact  the  friction  will  cause 
the  angle  of  separation  to  be  considerably  larger  than  45°; 
let  it  be  called  a.  Then,  in  order  that  there  maybe  perfect 
freedom  in  failure,  the  length  of  the  block  must  not  be  less 
than  its  least  width  or  breadth  multiplied  by  2  tan  a.  In 
some  cases,  a  has  been  found  to  be  about  55°,  for  which 
value. 

2  tan  a  =  2  X  1.43  =  2.86. 

If  the  bearing  faces  of  the  short  block  under  compres- 
sion are  of  much  area,  for  such  a  purpose,  it  will  be  difficult 
in  many  cases,  especially  with  large  loads,  to  secure  a 
uniform  application  of  those  loads.  The  resulting  ultimate 
resistance  for  the  entire  block  will  give  an  average  intensity 
of  pressure  which  may  be  quite  different  from  the  greatest 
intensity.  These  simple  considerations  are  particularly 
pertinent  to  such  materials  as  blocks  of  concrete  or  of 
natural  stone,  which  may  be  12  inches  square  or  more  in 
section. 

Again,  in  such  material  as  natural  or  artificial  stone  the 
friction  between  the  head  of  the  testing  machine  and  the 
bearing  surface  of  the  specimen,  or  along  the  planes  of 
greatest  ultimate  shear  will  tend  to  support  laterally  to 
some  extent  the  material  as  it  approaches  failure,  thus 
raising  the  apparent  ultimate  resistance  of  the  material. 
The  shorter  the  block  the  greater  will  be  this  frictional 
supporting  tendency.  This  effect  has  been  marked  where 
the  tests  specimens  have  been  cubes  varying  from  2  inches 
on  their  edges  to  12  inches,  the  large  cubes  showing  mate- 
rially greater  resisting  capacity. 


Failure  of  short  cylinders  of  cast  iron  showing  the 
shearing  of  the  metal  on  the  plane  of  maximum 
shear. 


View  exhibiting  the  failure  of  short  cylinders  of  Connecticut  brown  sandstone. 

(To  face  page  386.) 


Art.  63.]  WROUGHT  IRON.  3^7 

Art.  63. — Wrought  Iron. 

It  is  difficult  to  fix  the  point  of  failure  of  a  short  block 
of  wrought  iron  or  other  ductile  material.  As  the  load 
increases  above  the  elastic  limit,  the  cross-sections  of  the 
test  "piece  increase  in  lateral  dimensions  or  "  bulge  out," 
so  that  increase  of  compressive  force  simply  produces  an 
increased  area  of  resistance,  while  the  material  never  truly 
fails  by  crumbling  or  shearing  off  in  wedges. 

It  is  comparatively  easy  to  determine  the  elastic  limit, 
but  at  what  degree  of  loading  the  material  may  be  said  to 
fail  after  permanent  distortion  begins  is  not  clear  unless 
some  arbitrary  limit  should  be  fixed  by  convention. 

In  an  actual  structure  obviously  failure  may  be  said 
to  take  place  when  the  degree  of  distortion  is  such  that  the 
structure  fails  to  discharge  safely  its  function  as  a  load 
carrier,  but  that  degree  of  distortion  would  vary  much  in 
different  structures  or  in  different  parts,  possibly,  of  the 
same  structure. 

For  the  present  purpose  it  may  perhaps  be  assumed 
tentatively  that  a  ductile  material  fails  when  its  distortion 
under  compressive  loading  becomes  apparent  to  the  unaided 
eye. 

Modulus  of  Elasticity. 

As  wrought  iron  is  no  longer  a  structural  material,  there 
are  practically  no  recent  tests  to  determine  the  compressive 
modulus  of  elasticity,  but  earlier  investigators  made  suf- 
ficient tests  when  the  material  was  in  general  use  to  establish 
the  modulus  with  reasonable  accuracy.  Those  investi- 
gations show  that  there  is  no  essential  difference  between 
moduli  for  compression  and  tension.  Hence  the  modulus 
of  elasticity  for  wrought  iron  in  compression  may  be  taken 
at  26,000,000.  Small  specimens  would  in  some  cases  yield 


COMPRESSION.  [Ch.  VIII 

results  perhaps  as  high  as  28,000,000,  but  for  general  use 
the  former  or  smaller  value  is  preferable. 

Limit  of  Elasticity  and  Ultimate  Resistance. 

Investigations  for  determining  the  elastic  limit  of 
wrought  iron  in  compression  are  almost  entirely  lacking, 
but  its  value  may  safely  be  taken  the  same  as  for  tension, 
i.e.,  depending  upon  the  area  of  cross-section  and  the 
amount  of  work  put  upon  the  material  in  its  manufacture, 
from  22,000  to  perhaps  26,000  pounds  per  square  inch,  the 
former  for  large  sections  and  the  latter  for  small  sections. 
The  difficulties  met  in  the  effort  to  determine  a  well-defined 
ultimate  compressive  resistance  for  wrought  iron  have 
already  been  noticed,  but  such  compression  tests  as  were 
made  during  the  general  use  of  wrought  iron  for  structural 
purposes  indicate  that  what  may  be  termed  the  ultimate 
compressive  resistance  may  reasonably  be  taken  at  about 
the  ultimate  tensile  resistance.  The  amount  of  permanent 
distortion  taking  place  at  that  degree  of  loading  has  not 
been  satisfactorily  determined,  but  it  would  certainly  be 
apparent  to  the  unaided  eye  and  it  might  run  from  i  per 
cent,  to  5  per  cent,  or  possibly  more.  It  may  be  assumed, 
therefore,  that  the  ultimate  compressive  resistance  of 
wrought  iron  will  range  generally  from  45,000  to  50,000 
pounds  per  square  inch. 

Art.  64. — Cast  Iron. 

The  behavior  of  cast  iron  under  compression  as  found 
in  ordinary  casting  is  not  less  erratic  than  in  tension.  When 
this  material  was  used  for  such  purposes  as  heavy  ordnance 
and  car  wheels  it  was  so  produced  as  to  possess  excellent 
physical  qualities  for  a  cast  metal,  especially  after  remelting 
and  being  held  in  fusion.  Even  then,  however,  the  modulus 


Art.  64.]  CAST  fRON.  389 

of  elasticity  was  not  much  higher  than  for  the  best  qualities 
of  ordinary  castings.  It  may  be  said  generally  that  the 
modulus  of  elasticity  for  cast  iron  in  either  tension  or  com- 
pression may  be  taken  from  12,000,000  to  14,000,000. 
These  values  are  about  half  of  the  corresponding  values  for 
wrought  iron  and  little  less  than  half  the  corresponding 
values  for  structural  steel. 

Inasmuch  as  cast  iron  is  a  brittle  material  failing 
suddenly  at  the  limit  of  its  resisting  capacity,  either  in 
tension  or  compression,  it  can  scarcely  be  said  to  have  an 
elastic  limit  except  for  special  grades  of  unusual  excellence, 
and  even  with  such  material  it  is  not  well  defined. 

The  ultimate  resistance  of  cast  iron  to  compression  is 
fairly  well  defined, .but  it  varies  greatly  in  value  according 
to  its  quality.  Special  grades  for  ordnance  and  car  wheels 
may  have  compressive  resistances  running  from  100,000 
per  square  inch  up  to  150,000  pounds  per  square  inch. 
For  many  years  when  cast-iron  columns  were  used  in  engi- 
neering practice  it  was  customary  to  consider  the  ultimate 
compressive  resistance  for  such  members  as  100,000  pounds 
per  square  inch,  but  that  value  is  far  too  high.  Although 
the  quality  of  ordinary  castings  is  variable,  it  is  reasonable 
to  take  the  ultimate  compressive  resistance  at  80,000  pounds 
per  square  inch  for  such  material  as  may  be  used  under 
good  and  effective  specifications  for  columns,  machine 
frames  and  similar  purposes,  although  there  are  modern 
cast-iron  column  tests  which  appear  to  indicate  that  even 
that  value  is  too  high. 

Art.  65.— Steel. 

Table  I  of  Art.  58  contains  the  results  found  by  Prof. 
Ricketts  in  testing  cylindrical  specimens  of  mild  steel  in 
compression.  These  specimens  were  six  inches  long  be- 
tween carefully  faced  ends,  and,  as  the  table  shows,  their 


390  COMPRESSION.  [Ch.  VIII. 

diameter  was  about  0.75  inch.  The  coefficients  of 
elasticity  for  compression  were  found  by  measurements 
very  carefully  made  with  a  micrometer  on  a  length  of  four 
inches.  The  elastic  limits,  however,  were  determined  by 
operating  with  a  cylinder  two  inches  long,  and  were  taken 
at  those  points  where  the  material  of  the  specimens  ceased 
to  hold  up  the  scale  beam,  and  may  have  been  somewhat 
above  that  point  where  the  ratio  between  stress  and  strain 
ceases  to  be  essentially  constant. 

The  coefficients  of  elasticity  are  found  to  be  quite 
uniform,  irrespective  of  the  per  cents  of  carbon,  within  the 
limits  of  the  table,  and  they  are  seen  to  be  a  very  little 
less  than  the  coefficients  for  tension.  The  difference  is 
so  small  that  no  essential  error  will  arise  if,  for  all  en- 
gineering purposes,  they  are  assumed  the  same. 

A  comparison  of  the  elastic  limits  for  tension  and 
compression  presents  some  irregularities;  yet  with  the 
exception  of  the  high  percentages  of  carbon  in  the  last  two 
grades  of  Bessemer  metal,  the  two  sets  of  elastic  limits  as 
wholes  are  not  very  different  from  each  other.  In  the 
Bessemer  steel  with  the  two  high  per  cents  of  carbon,  the 
tensile  elastic  limits  are  materially  higher  than  those  for 
compression.  The  following  very  important  conclusion 
results  from  this  comparison  of  the  elastic  limits  for  the 
mild  structural  steels:  since  these  elastic  limits  are  es- 
sentially equal  it  is  not  only  permissible  but  wholly  rational 
to  increase  the  working  resistances  of  mild  steel  bridge 
columns  over  those  for  iron  in  at  least  the  same  proportion 
that  the  tensile  working  stress  of  the  same  steel  is  increased 
over  that  of  iron  in  tension.  Experiments  on  a  sufficient 
number  of  full-size  steel  columns  are  yet  lacking  to  verify 
this  conclusion. 

It  appears  from  such  data  on  the  compressive  resistance 
of  steel  as  exist  that  not  only  the  coefficient  of  elasticity 


Art.  65.]  STEEL.  39! 

but,  also,  the  limit  of  elasticity  in  compression  may  be 
taken  the  same  as  that  for  tension  for  the  same  grade  of 
steel.  This  was  practically  true  in  the  older  investiga- 
tions of  Kirkaldy,  and  it  is  essentially  confirmed  in  the 
few  later  investigations  available. 

The  ultimate  compressive  resistance  of  steel,  like  the 
ultimate  tensile  resistance,  varies  with  the  content  of 
carbon,  being  comparatively  low  with  a  small  percentage 
of  carbon,  and  correspondingly  large  with  a  high  percentage 
of  that  element.  It  is  also  much  affected  by  the  operations 
of  tempering  and  annealing. 

Special  grades  of  steel  adapted  to  heat  treatment  have 
after  such  treatment  given  ultimate  compressive  resistances 
of  various  values  up  to  nearly  or  quite  400,000  pounds  per 
square  inch  and  values  ranging  from  150,000  pounds  up 
to  300,000  pounds  per  square  inch  are  not  uncommon  in 
the  records  of  the  older  testing.  Such  high  results,  however, 
are  only  obtained  with  hardened  and  tempered  metal. 

There  is  the  same  uncertainty  as  to  the  point  at  which 
compressive  failure  takes  place  in  steel  which  attaches  to 
the  ultimate  compressive  resistance  of  all  ductile  metals 
and  which  was  commented  upon  in  Art.  63.  It  is  probably 
safe,  however,  if  not  entirely  correct,  to  take  the  ultimate 
compressive  resistances  of  different  grades  of  steel  equal 
to  their  ultimate  tensile  resistances  in  the  absence  of 
explicit  determinations;  and  a  similar  observation  may 
be  applied  to  the  working  resistances  in  pure  compression 
of  same  grades  of  steel. 

Art.  66.— Copper,  Tin,  Zinc,  Lead,  and  Alloys.* 

Table  I  shows  some  coefficients  of  elasticity  (i.e.,  ratios 
between  stress  and  strain),  computed  from  data  deter- 

*  As  this  field  of  investigation  has  not  been  worked  since  Prof.  Thurston 
left  it  his  results  are  allowed  to  stand  (1915). 


392 


COMPRESSION. 


[Ch.  VIII. 


mined  by  Prof.  Thurston,  and  given  by  him  in  the  "  Trans. 
Amer.  Soc.  of  Civ.  Engrs.,"  Sept.,  1881.  The  gun  bronze 
contained  copper,  89.97;  tin,  10.00;  flux,  0.03.  The  cast 
copper  was  cast  very  hot. 

TABLE  I. 


Coefficients  of  Elasticity  in 

Pounds  per  Square  Inch. 

Stress  in  Pounds 

per  Square  Inch. 

Gun  Bronze. 

Cast  Copper. 

1,620 



,254,000 

3,260 

3,622,000 

,415,000 

6,520 

4,075,000 

,651,000 

9,780 

6,II3,OOO 

,795,000 

13,040 

6,520,000 

,824,000 

16,300 

5,433,000 

,842,000 

19,560 

5,148,000 

,845,000 

22,820 

3,935,000 

,735,000 

26,080 

2,308,000 

1,503,000 

29,340 



1,144,000 

32,600 

1,073,000 

815,000 

48,900 

463,600 

332,500 

The  ratios  of  stress  over  strain  are  far  from  being  con- 
stant. Strictly  speaking,  therefore,  there  is  no  elastic 
limit  in  either  case.  In  that  of  the  gun  bronze,  however, 
it  may  be  approximately  taken  at  20,000  pounds  per  square 
inch  (Prof.  Thurston  takes  it  22,820),  and  in  that  of  the 
copper  at  25,000  pounds.  The  test  specimens  were  two 
inches  long  and  turned  to  0.625  inch  in  diameter. 

At  38,000  pounds  per  square  inch  the  gun  bronze  speci- 
men was  shortened  about  41  per  cent,  of  its  original  length, 
while  its  diameter  had  become  0.77  inch. 

The  copper  specimen  failed  at  71,700  pounds  per  square 
inch,  having  been  shortened  about  one  third  of  its  length. 

The  results  of  a  series  of  tests  by  Prof.  Thurston,  in 
connection  with  the  United  "States  testing  commission,  are 
given  in  Table  II;  they  were  abstracted  from  "  Mechanical 


Art.  66.]  COPPER,   TIN,  ZINC,  LEAD  AND  ALLOYS.  393 

TABLE  II. 


Composition. 

Pounds  per  Square  Inch 
Causing  a  Shortening  of 

Greatest 
Load  in 
Pounds 

Per  Cent, 
of  Short- 
ening 

Ultimate 
Crushing 
Resist- 

Manner of 

Caused 

ance  in 

Copper. 

Tin. 

5  Per 

Cent. 

10  Per 

Cent. 

20  Per 
Cent. 

per 

Square 
Inch. 

Greatest 
Load. 

Lbs.  per 
Sauare 
Inch. 

Failure. 

97-83 

1.92 

29,340 

34,000 

46,000 

46,260 

0.37 

34,000 

Flattened 

95.96 

3-80 

39,200 

42,050 

52,150 

52,150 

0.30 

42,050 

92.07 

7.76 

3i,5oo 

42,000 

65,000 

84,100 

0.45 

42,000 

90-43 

9-50 

32,000 

38,000 

60,000 

61,930 

0-34 

38,000 

87-15 

12.77 

39,000 

53,000 

80,000 

89,640 

0-39 

53,ooo 

80.99 
**  &  f\^ 

18.92 

65,000 

78,000 

103,490 

103,490 

0.  20 

78,000 

f*~**r^t*A 

70  .  DO 
69.90 

23-  23 
29.85 

101,040 

• 



1  1  4,080 
146,680 

o  .  09 

0.04 

I  T  4,080 
146,680 

L/rusneo. 

65.31 
61.83 

34-47 
37-74 



. 



84,750 
39,no 

o  .  03 
o  .  02 

84,750 

39,no 

47.72 

51-99 







84,750 

0.02 

84,750 

44.62 

55-15 







35,850 

0.01 

35,850 

38.83 

60.79 







39,110 

0.02 

39,no 

38.37 

61  .32 







29,340 

0.01 

29,340 

34.22 

65.80 

19,560 





I9,56o 

0.06 

I9,56o 

25-12 

74-51 

17,930 

17,930 

17,930 

17,930 

0.28 

17,930 

2O.  21 

79.62 

16,300 

16,300 

16,300 

16,300 

o.  29 

16,300 

15-12 

84-58 

6,520 

6,520 

6,520 

9,450 

0.51 

6,520 

Flattened 

11.48 

88.50 

10,100 

10,100 

10,100 

14,020 

0.50 

10,100 

8.57 

91-39 

6,500 





9,78o 

0.06 

9,780 

3-72 

96.  31 

6,520 

6,520 

6,520 

9,780 

0.34 

9,780 

0.74 

99.02 

6,520 

6,520 

6,520 

9,780 

0.36 

9,780 

0.32 

99.46 

6,520 

6,520 

6,520 

9,78o 

0.38 

9,780 

Cast  copper 

26,000 

39,000 

51,000 

74,970 

0.45 

39,000 

33,000 

45,500 

58,670 

78,230 

0.43 

45,5oo 

34,ooo 

42,000 

58,000 

71,710 

0.32 

42,000 

30,000 

36,000 

50,000 

104,300 

0.52 

36,000 

30,000 

37,000 

50,000 

91,270 

0.48 

37,ooo 

35,000 

48,000 

65,000 

97,790 

0.41 

48,000 

" 

Cast  tin 

6,030 

6,400 

6,530 

7,5oo 

0.44 

6,400 

and  Physical  Properties  of  the  Copper- tin  Alloys,"  United 
States  Report,  edited  by  Prof.  R.  H.  Thurston,  1879.  All 
the  specimens  were  0.625  inch  in  diameter  and  2  inches 
long.  Scarcely  one  of  them  can  be  said  to  possess  an 
elastic  limit. 

The  series  of  alloys  presents  some  interesting  results. 
About  the  middle  third  of  the  series  are  seen  to  be  brittle 
compounds  giving  (as  a  rule)  high  ultimate  compressive 
resistances,  while  the  other  two  thirds  are  ductile,  and  give 
at  the  copper  end  high  results,  and  low  ones  at  the  tin  end. 

It  will  be  observed  that  Prof.  Thurston  took  the  load  per 
square  inch  which  gave  a  shortening  of  10  per  cent,  of  the 
original  length  as  the  ultimate  resistance  to  crushing  of  the 


394 


COMPRESSION. 


[Ch.  VIII. 


ductile  alloys  and  metals,  since  such  materials  cannot  be 
said  to  completely  fail  under  any  pressure,  but  spread 
laterally  and  offer  increased  resistance. 

TABLE  III. 


Per  Cent,  of 

Pounds  per  Square  Inch  for 

Par   f*c»i-i4-      /-,-p 

Copper. 

Zinc. 

J&. 

Ultimate 
Resistance. 

er  i^ent.  01 
Shortening. 

Manner  of 
Failure. 

96.07 

3-79 

305,500 

29,000 

0.0 

Flattened 

90.56 

9.42 

342,100 

30,000 

10.  0 

89.80 

10.06 

29,500 

10.  0 

76.65 

23.08 

656,500 

42,000 

10.  0 

60.94 

38.65 

1,772,500 

75,000 

10.  0 

55-15 

44-44 

78,000 

10.  0 

49-66 

50.14 

1,345,500 

117,400 

10.  0 

47.56 

52.28 

I,5OO,OOO 

121,000 

10.  0 

25-77 

73-45 

4,232,800 

110,822 

5.85 

Crushed 

20.81 

77.63 

2,485,000 

52,152 

2-75 

14.19 

85.10 

897,000 

48,892 

10.8 

10.30 

88.88 



49,000 

IO.O 

Flattened 

4-35 

94-59 



48,000 

IO.O 

0.00 

IOO.OO 

318,500 

22,000 

IO.O 

Table  III  contains  the  results  of  Prof.  Thurston's  tests 
of  the  copper-zinc  alloys  made  while  he  was  a  member  of 
the  United  States  Board.  The  data  are  taken  from  "Ex. 
Doc.  23,  House  of  Representatives,  46th  Congress,  26. 
Session."  The  specimens  were  two  inches  long  and  0.625 
inch  in  diameter  of  circular  cross-section. 

The  values  of  E^  (ratios  of  stress  over  strain)  are  com- 
puted for  about  one  quarter  the  ultimate  resistance.  This 
ratio  is  so  very  variable  for  different  intensities  of  stress 
that  these  alloys  can  scarcely  be  said  to  have  a  proper 
"elastic  limit." 

Two  specimens  of  tobin  bronze,  each  .75  inch  in  diameter 
and  i  inch  long,  tested  by  the  Fairbanks  Company  of  New 
York  City  in  1891,  were  compressed  about  .8  per  cent,  at 
45,000  pounds  per  square  inch,  and  a  little  over  10  per  cent. 


Art.  67.]  CEMENT— CEMENT  MORTAR— CONCRETE.  395 

at  90,000  pounds  per  square  inch.  Tobin  bronze  contains 
58.2  per  cent,  copper,  2.3  per  cent,  tin,  and  39.5  per  cent, 
zinc. 

Art.  67.— Cement — Cement  Mortar — Concrete. 

The  ultimate  compressive  resistances  of  mortars  and 
concrete  determine  the  carrying  power  of  many  engineering 
works,  and  it  is  of  much  importance  to  ascertain  those 
resistances  and  the  conditions  under  which  they  may  be 
made  the  greatest  possible.  Obviously,  the  carrying  power 
in  compression  of  both  mortars  and  concretes  will  depend 
upon  a  considerable  number  of  elements  such  as  the  character 
of  the  cement,  the  proportions  of  mixture  of  the  sand  and 
cement  for  mortar  or  of  the  cement,  sand,  and  gravel  or 
broken  stone  for  concrete,  the  thoroughness  of  the  ad- 
mixture, the  amount  of  water  used,  the  conditions  under 
which  the  mortar  and  concrete  are  maintained  while  the 
operation  of  setting  is  taking  place,  the  temperature,  and 
other  various  influences. 

The  modulus  of  elasticity  of  concrete  must  necessarily 
depend  chiefly  upon  the  proportions  of  the  mixture  and  the 
age  of  the  concrete  when  tested.  It  will  also  depend  to  a 
material  extent  upon  the  intensity  of  compressive  stress  at 
which  the  strain  is  observed.  At  this  point  a  clear  under- 
standing of  the  elastic  behavior  of  the  mortars  and  concrete 
is  necessary  to  a  correspondingly  clear  view  of  what  takes 
place  in  a  concrete-steel  beam  under  loading.  In  many 
cases  of  concrete  under  compression  of  varying  intensities 
a  careful  measurement  of  the  resulting  strains  shows  that 
a  permanent  deformation  or  compression  remains  at  least 
for  the  time  being  after  the  removal  of  the  load,  even 
when  the  latter  is  sometimes  not  more  than  100  or  200 
pounds  per  square  inch.  This  permanent  set  is  dependent 
upon  the  age  of  the  material  and  usually,  perhaps  always, 


396  COMPRESSION.  [Ch.  VIII. 

decreases  as  age  increases.  In  many  other  cases  a  per- 
manent set  is  observable  only  under  intensities  of  stress 
'as  high  as  1000  or  1200  pounds  per  square  inch,  or  even 
considerably  more.  When  these  sets  occur  they  are  fre- 
quently found  far  below  what  may  probably  be  termed  the 
elastic  limit  of  the  material,  and  in  some  quarters  they  have 
given  the  impression  that  mortar  and  concrete  have  little 
or  no  true  elastic  behavior.  This,  however,  is  an  erroneous 
view,  as  in  the  testing  of  concrete  and  mortar  cubes  equal 
increments  of  stress  intensities  quite  uniformly  give  equal 
increments  of  strain  or  deformation  over  a  considerable 
range.  Although  the  upper  limit  of  this  essentially  constant 
ratio  between  stress  and  strain  is  usually  not  very  clearly 
defined,  it  is  so  .defined  in  a  considerable  percentage  of 
cases,  and  in  almost  all  tests  of  well-made  concrete  and 
mortar  that  limit  may  readily  be  assigned  near  enough  for 
all  practical  purposes. 

A  large  amount  of  data  bearing  upon  these  points  will 
be  found  in  the  "  Report  of  Tests  of  Metals  and  Other  Mate- 
rials" at  the  Watertown  Arsenal  for  1899.  Twelve-inch 
cubes  with  a  great  variety  of  proportions  of  constituent 
elements  ranging  from  a  few  days  up  to  six  months  in  age 
were  employed  in  those  investigations.  Figs,  i  and  2  ex- 
hibit graphically  the  results  of  twelve  of  those  tests  so  taken 
as  to  be  fairly  representative  of  all.  The  vertical  ordinates 
of  the  curves  represent  compressive  stress  intensities  up  to 
failure,  while  the  horizontal  ordinates  represent  the  total 
compressive  strains  or  deformation  under  the  corresponding 
stresses  also  up  to  the  point  of  failure.  These  strains  are 
shown  in  the  figures  one  hundred  times  their  actual  amounts. 
In  Fig.  i  the  concrete  nine  days  old  shows  only  little  resist- 
ing power  and  a  low  coefficient  of  elasticity,  as  would  be 
expected.  In  nearly  all  the  other  cases,  on  the  other  hand, 
the  ratio  between  stress  and  strain  is  reasonably  constant 


Art.  67.] 


CEMENT— CEMENT  MORTAR— CONCRETE. 


397 


up  to  nearly  1000  pounds  per  square  inch.  The  two  excep- 
tions are  found  in  Fig.  2,  belonging  to  i  to  3  Portland- 
cement  mortar  and  to  i,  2,  and  4  steel-cement  concrete,  the 
former  four  months  old  and  the  latter  three  months  old. 


2000  — 


1000 


On  the  other  hand,  the  1,2,  and  4  concrete  six  months  old 
in  the  right-hand  group  of  Fig.  i  discloses  constant  propor- 
tionality between  stress  and  strain  up  to  2000  pounds  per 
square  inch,  and  the  same  observation  may  apply  to  a  sim- 


398  COMPRESSION.  [Ch.  VIII. 

ilar  concrete  represented  by  one  of  the  curves  in  the  left- 
hand  group  of  Fig.  2.  Again  the  i  to  i  granite-dust  mortar 
four  months  old  represented  by  one  of  the  curves  in  the 
right-hand  group  of  Fig.  2  shows  a  constant  ratio  up  to 
nearly  4000  pounds  per  square  inch.  Indeed,  the  whole 
group  of  curves  probably  shows  a  more  satisfactory  approach 
to  a  constant  ratio  between  stress  and  strain  than  do  similar 
curves  for  cast  iron.  It  should  be  stated,  as  will  be  observed 
by  referring  to  the  report  cited,  that  some  of  the  curves 
shown  in  Fig.  i  and  Fig.  2  belong  to  groups  for  which  small 
permanent  sets  were  observed  below  elastic  limits,  while 
others  belong  to  those  which  show  no  such  permanent  set. 
This  observation  does  not  appear  from  the  test  records  to 
be  applicable  to  any  particular  character  of  curves,  but 
sometimes  to  those  which  are  more  nearly  straight  and  some- 
times to  those  which  are  less  so. 

The  results  deduced  from  the  tests  of  cubes  covered  by 
the  1899  and  other  "Reports  of  Tests  of  Metals  and  Other 
Materials"  are  confirmed  by  the  investigations  of  such  for- 
eign authorities  as  M.  Considere,  Melan,  Brik,  and  others. 
They  show  conclusively  that  it  is  reasonable  and  safe  to 
apply  to  concrete  and  concrete-steel  beams  the  formulae 
established  by  the  common  theory  of  flexure  after  intro- 
ducing into  them  empirical  quantities  established  by  experi- 
ment precisely  as  is  done  with  iron  and  steel  beams. 

Table  I  is  a  condensed  statement  of  average  values  of 
the  modulus  of  elasticity  for  concrete  of  different  propor- 
tions of  mixture  prepared  by  Mr.  Edwin  Thacher  from 
original  sources,  including  the  annual  Reports  of  Tests  of 
Metals  .and  Other  Materials  carried  on  by  U.  S.  officers  at 
the  Watertown  Arsenal  for  a  lecture  given  by  him  at  the 
College  of  Civil  Engineering  of  Cornell  University,  1902. 

This  table  exhibits  as  reasonable  values  for  the  coeffi- 
cient of  elasticity  in  compression  as  can  be  determined  at 


Art.  67.] 


CEMENT— CEMENT  MORTAR— CONCRETE. 


399 


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400  COMPRESSION.  [Ch.  VIII. 

the  present  time.  The  value  to  be  selected  for  any  particu- 
lar case  will  depend  upon  the  proportions  of  mixture  and 
upon  the  degree  of  balancing  of  the  sand  and  gravel  or 
broken  stone,  although  the  influence  of  the  latter  cannot 
be  definitely  stated.  It  is  not  improbable  that  a  considera- 
ble portion  at  least  of  the  variations  in  the  results  of  the 
table  are  due  to  the  varying  degrees  of  natural  balancing 
in  the  different  test  blocks.  The  value  will  also  depend 
upon  the  age  of  the  concrete.  For  all  ordinary  engineering 
constructions  it  is  reasonable  to  take  the  coefficient  of  com- 
pressive  elasticity  at  2,500,000  to  3,000,000  pounds  per 
square  inch  for  a  concrete  mixture  of  i  cement,  2  sand,  and 
4  gravel  or  broken  stone.  This  table  shows  that  practi- 
cally the  same  value  may  be  taken  for  a  concrete  of  i  cement, 
3  sand,  and  6  gravel  or  broken  stone,  especially  if  the  mate- 
rials are  well  selected  and  balanced.  If  the  concrete  is 
mixed  in  the  proportions  of  i  cement,  6  sand,  and  12  gravel 
or  broken  stone,  the  coefficient  of  elasticity  is  seen  to 
decrease  materially  and  should  not  be  taken  higher  than 
1,500,000  pounds  per  square  inch.  Suitable  quantities 
for  mixtures  other  than  those  named  in  the  table  can  be 
reasonably  and  safely  selected  from  those  afforded  in  it. 

These  values  show  that  the  ratio  of  the  coefficient  of 
elasticity  for  steel  over  that  for  concrete  may  range  from 
10  to  20  for  the  varying  conditions  described. 

The  more  common  practice  is  to  make  this  ratio  15,  i.e., 
on  the  basis  of  30,000,000,  for  the  modulus  of  elasticity  for 
steel  and  2,000,000  for  concrete.  The  ratio  of  12,  however, 
is  sometimes  found  by  taking  the  same  value  as  before  for 
the  modulus  of  steel,  but  2,500,000  for  the  modulus  of 
elasticity  for  concrete.  The  ratio  of  the  two  moduli  is 
constantly  used  in  the  treatment  of  reinforced  concrete 
work. 

A  further  consideration  must  be  kept  in  view  in  con- 


Art.  67.] 


CEMENT— CEMENT  MORTAR— CONCRETE. 


401 


nection  with  the  value  of  the  modulus  of  elasticity  for  con- 
crete, and  that  is  the  fact  alluded  to  in  previous  pages  that 
nearly  all  concrete  and  reinforced  concrete  work  must 
usually  carry  considerable  loading,  in  the  exigencies  of  con- 
struction, when  it  has  attained  no  greater  .age  than  perhaps 
i o  to  30  days,  i.e.,  before  the  modulus  of  elasticity  (or  ultimate 
resistance)  has  attained  its  full  value.  Again,  a  large  mass 
of  concrete,  as  actually  built,  cannot  reasonably  be  expected 
to  have  as  high  a  modulus  as  12 -inch  cubes  or  other  com- 
paratively small  pieces  made  and  tested  in  a  laboratory. 
For  all  these  reasons  it  is  prudent  to  take  a  rather  low  value 
of  the  modulus  of  elasticity  for  the  analytic  work  of  design. 
The  following  tabulated  statement  shows  ultimate  resist- 
ances per  square  inch  of  12 -inch  cubes  of  concrete  obtained 
in  the  Testing  Laboratory  of  the  Department  of  Civil 
Engineering  of  Columbia  University  in  1912  by  Mr.  James 
S.  Macgregor,  in  charge  of  the  laboratory. 

GRAVEL  CONCRETE;    i  Cement,  2%  Sand,  5  Gravel. 


Ult.  Resistance  Pounds  per  Sq.  In. 

Max. 

Mean. 

Min. 

Alsen          

I.9I7 
1,905 
2,223 
1,789* 
1,848 
2,717 
2,278 
I,l62* 

i,735 
2,322 

1,202 

1.773 
1,796 
2,191 
i,553* 
i,778 
2,584 
2,139 
1,097* 

i,593 
2,088 
1,023 

i,557 
1,706 

2,152 
1,431* 
1,657 
2,431 
2,007 

1,047* 
1,518 

2,006 
944 

Age  of 
all 
cubes 
42 

days 

Atlas          

Atlas        

Iron  Clad  

Iron  Clad 

Lehigh 

Lehigh 

Vulcanite 

Vulcanite               

Alsen  
Alsen*  

*  Gravel  unwashed. 


The  coarse  aggregate  for  all  cubes  was  river  gravel  with 
stones  up  to  i-inch  size.  Some  of  the  gravel  contained  an 
excessive  amount  of  dirt  or  other  fine  material,  which 


402 


COMPRESSION, 


[Ch.  VIII. 


TABLE  II. 

MEAN  ULTIMATE  COMPRESSIVE  RESISTANCES  OF  12-INCH  PORT- 
LAND-CEMENT CONCRETE  CUBES. 


Mean  Ultimate  Resistance, 

Coefficient  of  Elasticity  in 

Portland  Cements. 
Brand;  Composition. 

Pounds  per  Square  Inch 
at  Age. 

Pounds  per  Square  Inch 
at  Age. 

7  Days. 

i  Mo. 

3  Mos. 

6  Mos. 

i  Mo. 

3  Mos. 

6  Mos. 

(      c.,  2  s.,  4  b.  st. 

1,724 

2,238 

2,702 

3,5o6 

2,500,000 

3,571,000 

5,000,000 

Saylors.  .  X      c.,  3  s.    6  b.  st. 

1,625 

2,568 

2,882 

3,567 

2,778,000 

4,167,006 

2,500,000 

(      c.,  6  s.  12  b.st. 

675 

800 

1,128 

1,542 

833.000 

2,273,000 

2,083,000 

I         C.,   2  S.     4  b.  St. 

1,387 

2,428 

2,966 

3,953 

3,1  25,000 

4,167,000 

3,125,000 

Atlas  ...-•<      c.,  3  s.   6b.  st. 

1,050 

1,816 

2,538 

3,170 

3,125,000 

2,778,000 

3,571,000 

(      c.,  6  s.  12  b.st. 

594 

1,090 

1,201 

1,583 

1,316,000 

1,136,000 

1,786,000 

f        C.,  OS.     2  b.  St. 

3,294 

5,053 

5,047 

3,125,000 

5,000,000 



Al^lia         J      c->  2  s-   4  b.  st. 

902 

2,420 

3,123 

4,411 

2,083,000 

4,167,000 

3,125,000 

Alpha.  .  .  *      c.;  3  s.   6  b.  st. 

892 

5,150 

2,355 

2,750 

2,083,000 

3,571,000 

4,167,000 

t     c.,  6  s.  12  b.st. 

564 

i,  218 

1,257 

1,532 

1,667,000 

1,786,000 

1,923,000 

f       C.,  OS.      2  b.    St. 

2,734 

3,246 

3-858 

5,129 

3,571,000 

2,778,000 

3,571,000 

Germania-j     £''  ^  |'   £  b'  |£' 

2,219 

1,550 

2,642 
2,174 

3,082 
2,486 

3,643 
2,930 

2,273,000 

3,571,000 

2,778,000 

4,167,000 
3,1  25,000 

t     c.',  6  s.  12  b.  st. 

759 

987 

963 

815 

961,000 

2,083,000 

1,786,000 

f       C.,  OS.     2  b.    St. 

3,240 

3,7io 

5,332 

2,273,000 

2,273,000 

3,571,000 

Alvn            ^       Cl>  2  S>    4  b"   St" 

1,592 

2,269 

2,608 

3,612 

2,788,000 

2,778,000 

4,167,000 

A1S6n  '  '  '  1      c.,  3  s.   6  b.  st 

2,114 

2,349 

3,026 

2,273,000 

2,778,000 

3,571,000 

I     c.,  6s.  ,12  b.  st 

'417 

873 

844 

1,323 

i  ,562,000 

i  ,562,000 

1,786,000 

10-INCH    CUBES. 


Alpha.  .  .     i  c.,  o  s.,  2  b.  st. 

5,463 

6,556 

5,000,000 

In  this  table  each  ultimate  resistance  is  a  mean  of  four  to  six  tests. 


TABLE  III. 

MEAN  ULTIMATE  COMPRESSIVE  RESISTANCES  OF  12-INCH  PORT- 
LAND-CEMENT CONCRETE  CUBES  WITH  LOAD  TAKEN  ON 
8"  BY  8".25  PLATE  ON  ONE  FACE. 


Mean  Ultimate  Resistance, 

Pounds  per  Square  Inch 

Portland  Cements. 

at  Age. 

Brand; 

Composition. 

i  Month. 

3  Months. 

6  Months. 

Alpha....  | 

C.,  O  S.,  2  b.  St... 
"  2  "  4  "  ... 

5,089 

3,287 

4,531 

Each  ultimate 
resistance     is     a 

5,669 

Germania  j 

C.,  O  S.,  2  b.  St... 

"  2  "  4  "  ... 

4,327 
3,587 

3,522 

6,671 

4,582 

mean     of 
tests. 

three 

Alsen  ....  | 

C.,  0  S.,  2  b.  St... 
"  2  "  4 

4,087 
3,233 

3,426 

6,382 
4,983 

A  view  exhibiting  the  failure  under  compression  of  a  12-in.  concrete  cube.  The 
composition  is  I  Portland  cement,  I  sand,  and  4.5  broken  stone.  The  age  of 
the  concrete  was  I  year,  8  months,  23  days,  and  the  ultimate  compressive 
resistance  attained  was  4481  Ibs.  per  sq.  in. 

(To  face  page  402.) 


Art.  67.]  CEMENT— CEMENT  MORTAR— CONCRETE.  403 

accounts  for  the  low  values  of  the  starred  ultimate  resist- 
ances per  square  inch,  as  indicated  by  the  footnote.  The 
age  of  all  the  cubes  was  42  days,  also  as  indicated  in  the 
table.  These  results  are  unusually  valuable  in  one  respect, 
in  that  the  cubes  were  not  mixed  in  the  laboratory,  but  in 
the  field,  where  actual  work  was  being  done,  and  hence 
received  no  special  care  in  the  operation. 

Tables  II  and  III  contain  the  results  taken  from  the 
"  U.  S.  Report  of  Tests  of  Metals  and  Other  Materials  "  for 
1899.  They  exhibit  the  ultimate  compressive  resistances 
of  cubes  of  Portland-cement  concrete,  the  cements  being 
among  the  well-known  brands.  The  ages  of  these  cubes 
vary  from  seven  days  to  six  months.  The  data  show 
clearly  the  increase  of  ultimate  resistance  with  the  ages  of 
the  cubes,  and  the  same  observation  applies  to  the  three 
columns  showing  the  coefficients  of  elasticity  at  one  month, 
three  months,  and  six  months.  The  compositions  of  the 
different  concretes  of  Table  II  are  those  quite  generally 
employed  in  engineering  practice. 

Table  III  exhibits  the  ultimate  resistances  of  the  same 
concretes,  but  with  the  pressure  applied  to  the  1 2-inch 
cubes  on  areas  8  inches  by  8J  inches,  this  end  being  at- 
tained by  the  use  of  steel  plates.  As  would  be  expected, 
the  ultimate  resistances  are  seen  to  be  considerably  greater 
than  are  found  with  the  total  load  distributed  over  the 
entire  surface  of  a  cube. 

The  broken  stone  used  in  the  cubes,  the  results  of  whose 
tests  are  given  in  Tables  II  and  III,  was  a  conglomerate  from 
Roxbury,  Mass.,  and  the  sand  was  coarse,  clean,  and  sharp. 
The  voids  of  the  broken  stone  measured  49.5  per  cent,  of 
their  total  volume. 

Table  IV,  taken  from  the  same  volume  of  the  "  U.  S. 
Report  of  Tests  of  Metal  and  Other  Materials  "as  Tables  II 
and  III,  exhibits  the  ultimate  compressive  resistances  of 


404 


COMPRESSION. 


[Ch.  VIII. 


TABLE  IV. 

MEAN  ULTIMATE  COMPRESSIVE  RESISTANCES  OF  MORTAR  AND 
CONCRETE  12-INCH  CUBES. 


Brand;    Composition. 


Mean  Ultimate 

Resistance, 

Pounds  per 

Square  Inch 

at  Age  of 

Four  Months. 


Weight  per 

Cubic  Foot, 

Pounds. 


Coefficient  of 

Elasticity, 

Pounds  per 

Square  Inch. 


f     c.,  i  s.,  ob.  st 4,371 

.  "      2  "     O  "      2,506 

*st«  \-\-l  IE  "S 

I  i  "    6  "    o  "    185 

LI"    7  "    o  "    118 

eind      1  '  "    '  "    °  "    • 

Portland       ['"    '"   °  "    5.°« 

fgSSd'      1'"    <"°  "    3,979 

PoZnt     ['"    '"°  "    4,353 

Poland       }'"    '*     °  "    5,306 

Steel  slag        i  "    i  s.,  o  "    i,743 

i  "    2  "    4  '    1,939 1 

Hoffman       )     «,       ,,  « 
Rosendale    J  J 

Norton          j  T  i',    x  ",    °  ""'"'•  6^  , 

Rosendale    |{,,    22 ,   «  ,,;;;;;  g-| 


136.5 
134-2 
133-8 
120.9 

"9-5 
116.9 
111.5 

141-5 

134-5 
134-7 
134-7 
137-3 

126.6 
152.1 

127.7 

125.2 
120.7 
146.2 


3,571,000 
3,125,000 
1,786,000 


6,250,000 
4,167,000 
3,125,000 
2,500,000 
3,571,000 

1,190,000 
2,5OO,OOO 


*  Granite  dust. 


f  Age,  3  months. 


J  Trap  rock,  broken  stone. 


TABLE  V. 

CHEMICAL  ANALYSES  OF  PORTLAND  AND  STEEL-SLAG  CEMENTS. 


Cement. 


Silica. 


Oxide 
of  Iron. 


Alumina. 


Lime. 


Magnesia. 


Sulphur 
Trioxide. 


Carbon 
Dioxide. 


Alpha.  .  . 
Star.  .  .  . 
Standard 
Alsen.  .  . 
Steel.  . 


20 

21.73 

22.5 

20.67 

31.02 


2.8 
2-5 

2.6 
2  .  I 

Trace 


10.87 

9-47 
11.98 
14.6 
10.9 


58.66 
56.34 
51-44 
42 . 16 

57-31 


3-35 
3.61 
3.6i 
2.32 
4-05 


1-34 
1.91 

1-57 
2.32 
3.36 


2.56 
3-94 
5.96 

4-45 
4.81 


Art.  67.] 


CEMENT— CEMENT  MORTAR— CONCRETE. 


405 


the  mortar  and  concrete  1 2-inch  cubes  described  therein. 
These  results  need  no  explanation,  as  they  are  similar  to 
those  which  have  already  been  given,  but  it  is  well  to  note 
that  the  last  four  lines  of  the  table  give  results  belonging  to 
two  brands  of  natural  cement.  There  are  also  shown  one 
test  of  a  steel-slag  cement  mortar  cube  and  one  of  concrete. 
Table  V  exhibits  the  chemical  analyses  of  the  Portland 
and  steel-slag  cements  named  in  Table  IV.  These  analyses 
exhibit  about  the  usual  composition  of  the  various  grades 
of  cement  to  which  they  belong. 

TABLE  VI. 

COMPRESSION  TESTS  OF  12-INCH  CUBES  OF  PORTLAND-CEMENT 
CINDER  CONCRETE. 


Brand. 

Composition. 

Age 
when 
Tested, 
Days.  . 

No.  Cubes 
Tested. 

£*$ 
fw5 

§ 

Ultimate  Resistance 
in  Lbs.  per  Sq.  In. 

Coefficient 
of 
Elasticity, 
Pounds. 

Max. 

Mean. 

Least. 

Germania  . 

Alpha  ". 
Atlas.'.'.'.'  '. 

c     i  s.   3  cinder 
c     2  s.   3 
c     2  s.   4 

C       2  S.     5 

c     3  s.   6 
c     i  s.    3 
c     2  s.    5 
c.    is.    3 

C.     2  S.    -5 

99  and  102 

IO2 

98 
98  and  101 
9i 
90 
90 
90 
90 

3 

3 
3 
3 
3 
3 
3 
3 
3 

110.4 

112.  8 

107.9 

106.3 

103.5 
114.1 

no 
116.3 
109.9 

2,023 
i  ,701 
1,344 
1,114 
854 
2,988 
1,715 
2,580 
1,263 

2,001 

1,634 
1,325 
1,084 
788 
2,834 
i,  600 
2,414 
1,223 

,975 
,589 
,295 
,052 
749 
,780 
,402 
.295 

,200 

—  :  — 

2,500,000 
1,279,000 
3,125,000 
857,000 

The  results  exhibited  in  Table  VI  are  interesting  as 
belonging  to  Portland-cement  cinder  concrete  and  they  are 
of  practical  importance  because  such  concrete  is  used  in 
many  buildings  especially  for  floors,  in  consequence  of  its 
weighing  much  less  than  ordinary  broken-stone  concrete. 
The  ages  of  these  cinder  concrete  cubes  is  seen  to  run  from 
90  to  102  days,  which  is  sufficient  to  give  nearly  the  full 
ultimate  resistance  of  such  material.  It  is  seen,  however, 
that  cinder  concrete  is  materially  less  strong  or  capable  of 
ultimate  compressive  resistance  than  either  broken-stone 
or  gravel  concrete  having  the  same  proportions  of  mixture 
in  rts  composition.  The  column  giving  the  weight  in 


406 


COMPRESSION. 


[Ch.  VIII. 


pounds  per  cubic  foot  shows  that  cinder  concrete  weighs 
but  about  three  fourths  as  much  as  that  made  with  gravel 
and  broken  stone.  The  data  contained  in  this  table  were 
taken  from  the  "  U.  S.  Report  of  Tests  of  Metal  and 'Other 
Materials"  for  1898. 

Messrs.  Harold  Perrine,  C.E.  and  George  E.  Str^nan, 
C.E.  presented  a  paper  to  the  Am.  Soc.  C.  E.  in  1915 
describing  their  extended  investigation*  in  "  Cinder  Con- 
crete for  Floor  Construction  between  Steel  Beams."  The 
Table  VII  is  taken  from  that  paper  and  each  value  is  a 
mean  of  ten  results,  except  those  in  the  second  column 

TABLE   VII. 


Proportions  

Method 

C.  S.  Cin. 
1:2:5 

Continuous 
mixer. 
Coltrin. 
Alsen. 

C.  S.  Cin. 
1:1:5 
By  hand 
turned 
twice. 
Dragon. 

C.  S.  Cin. 
1:2:5 

Batch 
mixer. 

Vulcanite. 

C.  S.  Cin. 
1:2:5 
Ransome. 
Mixer, 

Atlas. 

Cement  

Sand  

Long  Island  Bank  Sand,  North  Shore. 

Cinders. 

Anthracite. 

.Ice 
plant. 

Local 
hotel 
steam 
plant. 

Local. 

Local  office 
building 
steam 
plant. 

Weight  Ibs  per  cu  ft 

107 

407 
924,600 

701 
I,I34,OOO 

933 
971,000 

913 
993,000 

100 

507 
857,400 

662 
I,O3O,OOO 

754 
1,050,000 

813 
956,000 

107 

818 
1,230,000 

1,254 
1,740,000 

1,744 
1,348,000 

1465 
1,200,000 

I€>9 

980 
1,492,000 

1,035 

1,428,250 

1,478 
1,276,000 

i,475 
1,320,000 

One  month  test: 
Ult.  Resist.,  Ibs.  per  sq.  in. 
E,  Ibs.  per  sq.  in  
Two  months  test: 
Ult.  Resist.,  Ibs.  per  sq.  in. 
E,  Ibs.  per  sq.  in  
Six  months  test: 
Ult.  Resist.,  Ibs.  per  sq.  in. 
E  Ibs  per  sq.  in      

One  year  test: 
Ult.  Resist.,  Ibs.  per  sq.  in. 
E,  Ibs.  per  sq.  in  

*  Made  in  the  testing  laboratory  of  the  Dept.  of  Civil  Engineering,  Col- 
umbia University  by  the  aid  of  the  Wm.  R.  Peters,  Jr.  memorial  research  fund. 


Art.  67.]  CEMENT— CEMENT  MORTAR— CONCRETE.  407 

from  the  right  side  of  the  Table,  which  are  means  of  nearly 
that  number.  The  compressive  test  specimens  were  cinder- 
concrete  cylinders  8  inches  in  diameter  and  16  inches  long. 
The  values  given  in  the  Table  are  representative  of  good 
structural  cinder  concrete. 

A  large  number  of  tests,  the  results  of  which  need  not 
be  given  here,  have  shown  that  gravel  may  advantageously 
be  used,  in  the  interests  of  economy,  in  the  place  of  broken 
stone  for  concrete.  On  the  whole,  the  broken-stone  concrete 
is  probably  stronger  than  that  made  with  gravel,  but  the 
difference  is  not  material  for  all  ordinary  cases.  The 
gravel  should  not  be  water-worn,  but  have  sharp,  gritty 
surfaces  to  which  the  setting  cement  may  strongly  bond 
itself.  All  sizes  from  the  largest  permissible  down  to 
coarse  sand  should  be  taken,  and  when  so  balanced  the 
voids  may  be  reduced  as  low  as  20  per  cent,  of  the  total 
volume  of  the  gravel  or  even  lower.  This  balancing 
of  the  broken  stone  or  gravel  enhances  both  economy 
and  resisting  qualities. 

A  careful  examination  of  all  the  Tables,  I  to  V,  shows 
that  reasonably  well-made  broken-stone  concrete  may 
carry  a  load  of  300  to  500  pounds  per  square  inch  without 
exceeding  \  to  J,  or  possibly  -J,  of  its  ultimate  resistance, 
the  composition  of  the  mixture  being  i  cement,  2  sand, 
and  4  broken  stone,  or  perhaps  i  cement,  3  sand,  and  5 
broken  stone.  It  is  possible  that  this  may  be  an  under 
statement  of  the  capacity  of  the  concrete  if  the  mixture  is  as 
well  balanced  as  it  should  be.  It  is  a  mistake,  as  has  been 
shown  repeatedly  by  actual  test,  to  screen  out  the  finer 
portions  of  the  broken  stone  or  to  attempt  to  secure  an 
approximately  even  sand  grain.  It  is  conducive  to  an 
increased  resistance  as  it  is  to  increased  economy  to  balance 
the  sand,  gravel,  or  broken  stone  by  using  all  the  varying 
sizes  between  the  least  and  the  greatest.  Indeed,  in  many 


408 


COMPRESSION. 


[Ch.  VIII. 


TABLE   VIII. 
COMPRESSIVE  RESISTANCES  OF  12"  x  12"   CONCRETE    COLUMNS. 


11 

Age, 
Days. 

Composition. 

W'ght 
n  Lbs. 
per 
Cu.Ft. 

Ult. 
Resist,  in 
Lbs.  per 
Sq.  In. 

2 

477] 

i    cement,   3   sand,    4-1  \"  } 
broken  stone,  2—  £"  broken  \ 

145 

1,072] 

2 

47] 

stone                                       ) 

145 

917 

4 

47 

do. 

144 

1,067 

4 

47 

do. 

144 

1,132 

6 

46 

do. 

844 

6 

46 

do. 

143 

1,048 

8 

42 

do. 

145 

935    - 

Hand  mixed 

8 

42 

do. 

145 

900 

10 

40 

do. 

142 

909 

IO 

4i 

do. 

H3 

807 

12 

39 

do. 

144 

947 

12 

39 

do. 

144 

980 

14 

34 

do. 

145 

936 

14 

35 

do. 

145 

907  , 

2 
2 

47  j 
47  ( 

i    cement,  3   gravel,  4-1  i'') 
broken  stone,  2—  f  "  broken  >• 
stone                                       ) 

145 
H7 

1,185 
1,183 

4 

48 

do. 

143 

980. 

4 

48 

do. 

144 

936 

6 

48 

do. 

146 

1,131 

6 

48 

do. 

146 

,200 

8 

42 

do. 

146 

,108 

8 

42 

do. 

146 

,086 

10 

do. 

146 

,015 

10 

42 

do. 

146 

,000 

12 

37 

do. 

149 

1,400 

12 

39 

do. 

148 

1,500 

14 

35 

do. 

148 

858 

14 

6 
6 

35 
4.1 

42  1 

do. 
i  cement,  6  gravel,  8-i£"  ) 
broken  stone,  \-\"  broken  \ 
stone                                        ) 

148 

143 
144 

807 

500 
467 

Machine  mixed 

6 
6 

42  \ 
42  1 

i  cement,  7  gravel,  8§-i^"  ) 
br'k'n  stone,  4*-f"  br'k'n  \ 
stone 

141 

142 

427 
436 

«( 

i  cement,  5  gravel,  6§-i£"  ) 

146 

708 

451 

y|  C     I 

br'k'n  stone,  3^-f"  br'k'n  \ 

146 

747 

45  j 

stone                                      ) 

6 
6 

46  f 

A6    1 

i   cement,  4  gravel,  5i-ii"  ) 
br'k'n  stone,  2§-f"  br'k'n  [• 

146 

145 

900 

797 

T-U  1 

stone                                      ) 

12 
12 

36  ( 

39  1 

i    cement,   3  gravel,   6-f  "  j_ 
broken  stone                         ) 

150 
149 

1,250 
1,700  J 

1  Reinforced    with 
!    4-i"    cold-twisted 

j    steel  rods  embed- 

J  ded  in  the  concrete 

( 

i  Silica  Portland  cement,  2  \ 

9 

58o-^ 

coarse  clean  sand,  3  quartz  V 

148 

2,548 

gravel  ($"-2")                      i 

Art.  68.]  BRICKS  AND  BRICK  PIERS.  409 

cases  it  may  be  advisable  to  use  the  entire  product  of  the 
crusher. 

The  relation  between  the  ultimate  compressive  resist- 
ance of  concrete  made  with  balanced  material  and  the 
length  of  column  is  illustrated  by  the  results  given  in  Table 
VIII,  which  has  been  collated  and  arranged  from  the  "  U.  S. 
Report  of  Tests  of  Metal  and  Other  Materials  "for  1897. 
The  heights  of  column  range  from  2  to  14  feet.  While  there 
are  some  exceptions,  the  rule  is  general  that,  other  things 
being  equal,  the  ultimate  resistance  decreases  as  the  length 
or  height  of  column  increases.  On  the  whole,  the  machine- 
mixed  material  appears  to  be  a  little  stronger  than  the 
hand-mixed,  but  the  difference  is  not  substantial  except  for 
the  8,  10,  and  12  feet  lengths. 

Art.  68. — Bricks  and  Brick  Piers. 

The  ultimate  compressive  resistance  of  bricks  depends 
largely  upon  the  manner  in  which  they  are  tested  and  the 
care  with  which  the  surfaces  pressed  are  filled  out  with  a 
proper  cushion  and  made  truly  parallel  to  the  bearing 
surfaces  of  the  testing  machine.  The  best  of  bricks  as 
produced  for  the  market  do  not  have  opposite  faces  truly 
parallel,  and  hence  when  they  are  placed  in  a  testing 
machine  for  testing  to  failure  the  pressure  will  be  con- 
centrated at  different  points  and  the  bricks  will  be  broken 
partly  by  bending  before  the  full  ultimate  compressive 
resistance  is  developed  unless  the  pressed  surfaces  are 
made  true  by  some  kind  of  a  cushion.  This  cushioning  is 
frequently  and  perhaps  usually  done  with  plaster  of  paris, 
as  in  the  case  of  the  tests  of  bricks  at  the  U.  S.  Arsenal, 
Watertown,  Mass.,  the  results  of  which  are  given  in  Table  II. 
Again,  a  brick  tested  on  edge  will  give  a  less  ultimate 
resistance  per  square  inch  than  when  tested  flat  and  the 


410  COMPRESSION.  [Ch.  VIII. 

resistance  on  end  per  square  inch  of  section  will  be  less 
than  that  on  edge.  When  the  brick  is  tested  flatwise, 
even  when  truly  surfaced  with  a  cushion  such  as  plaster  of 
paris,  it  is  a  very  short  block  and  the  friction  of  the  pressed 
surfaces  on  the  bearing  faces  of  the  testing  machine  is 
sufficient  to  give  the  compressed  material  substantial  lat- 
eral support,  not  permitting  it  to  separate  and  crush  away 
readily.  It  will  be  found,  therefore,  that  when  blocks  are 
tested  flatwise  the  ultimate  resistances  per  square  inch, 
as  a  whole,  will  be  much  higher  than  when  tested  on  edge. 
This  condition  of  things  holds  to  some  extent  when  the 
bricks  are  tested  on  edge,  so  that  an  endwise  test  will  give 
the  ultimate  compressive  resistance  per  square  inch  some- 
what less  than  that  found  when  the  brick  is  tested  on  edge. 
An  endwise  test  of  the  brick  more  truly  represents  the 
ultimate  compressive  resistance  of  the  material  than  a  test 
either  flatwise  or  on  edge. 

A  series  of  tests  of  a  variety  of  bricks  and  terra-cotta 
made  in  1896  at  the  U.  S.  Arsenal  at  Watertown,  Mass., 
gave  moduli  of  elasticity  about  as  follows:  Pressed  brick, 
1,000,000  to  3,000,000  pounds  per  square  inch,  the  hardest 
varieties  giving  the  higher  values  and  the  softer  material, 
the  lower  values;  hard  buff  brick  and  terra-cotta,  4,000,000 
to  4,800,000  pounds  per  square  inch.  Some  soft-face  brick 
gave  moduli  of  elasticity  varying  from  about  400,000  to 
890,000  pounds  per  square  inch.  These  determinations  of 
the  modulus  were  made  with  intensities  of  pressure  from 
about  1000  to  4000  or  5000  pounds  per  square  inch. 
Such  experimental  results  ordinarily  show  some  erratic  or 
abnormal  features  and  these  tests  were  no  exception  to 
that  rule. 

The  coefficients  of  thermal  expansion  and  contraction 
per  degree  Fahr.,  were  at  the  same  time  found  to  range 
from  .00000205  to  .00000754,  the  larger  of  these  values 


A  solid  i6-inch  square-face  brick  pier  laid  in  lime 
mortar  It  was  tested  at  the  U.  S.  Arsenal,  Water- 
town,  Mass.,  and  gave  an  ultimate  compressive 
resistance  of  1337  Ibs.  per  sq.  in.  The  pier  is 
shown  as  it  existed  after  failure. 

(To  face  page  410.) 


Art.  68.1 


BRICKS  AND   BRICK  PIERS. 


411 


being  about   25   per  cent,   higher  than  the  coefficient  for 
concrete. 

In  the  Proceedings  of  the  Am.  Soc.  C.  E.  for  March, 
1903,  Mr.  S.  M.  Turrill,  Assoc.  Am.  Soc.  C.  E.,  gives  the 
results  of  a  large  number  of  tests  of  common  building 
brick,  2  in.  by  4  in.  by  8  in.  in  size,  manufactured  at  Horse- 
heads,  N.  Y.  The  following  table  is  fairly  representative 
of  the  results  of  Mr.  Turrill's  tests,  made  with  great  care 
at  the  civil -engineering  laboratories  of  Cornell  University: 

TEST   OF   COMMON   BUILDING   BRICK. 


Brick  Tested. 

No.  of  Tests. 

Ultimate  Compressive  Resistance, 
Pounds  per  Square  Inch. 

Greatest. 

Mean. 

Least. 

On  end                

12 
12 
12 

3,763 
3,913 
5,463 

2,628 
2,832 
3,995 

1,234 
1,897 
2,665 

On  edge     

Flat  

These  bricks  were  tested  in  their  natural  condition  as 
delivered  from  the  kiln  ready  for  use. 

Other  tests  were  made  of  the  same  brick  saturated  with 
water  and  after  being  reheated  in  a  suitable  oven.  This 
latter  test  was  designed  to  disclose  the  quality  of  brick 
after  having  passed  through  a  conflagration.  The  satu- 
rated bricks  tested  on  end  and  on  edge  showed  material 
loss  of  resistance  below  that  of  their  natural  condition,  but 
those  tested  flat  showed  large  gains.  The  reheated  bricks 
exhibited  large  gains  in  all  three  modes  of  testing.  These 
bricks  were  obviously  not  of  hard-burned,  high-resisting 
character. 

The  coefficient  of  elasticity  of  twelve  of  these  bricks 
ran  from  540,000  to  1,815,000  pounds  per  square  inch,  with 
a  mean  value  of  1,305,000  pounds. 


412 


COMPRESSION. 


[Ch.  VIII. 


A  large  number  of  determinations  of  the  ultimate  com- 
pressive  resistances  of  bricks  were  made  among  the  earlier 
experimental  investigations  at  the  U.  S.  Arsenal  at  Water- 
town,  Mass.  These  results  showed  values  for  hard-burned 
bricks  varying  from  about  8,000  to  about  12,000  pounds 
per  square  inch  with  an  average  of  about  9,000  pounds  per 
square  inch  when  tested  on  edge.  What  may  be  termed 
medium  bricks,  i.e.,  intermediate  between  hard-burned 
strongest  bricks  and  common  building  bricks,  gave  results 
varying  from  about  4,000  to  about  8,000  pounds  per  square 
inch,  with  an  average  value  of  about  5,500  pounds  per  square 
inch  when  tested  on  edge. 

The  following  results  of  tests  of  three  different  kind 
of  brick  and  hollow  tile  were  obtained  by  Mr.  J.  S.  Mac- 
gregor  in  the  testing  laboratory  of  the  Department  of  Civil 
Engineering  at  Columbia  University.  The  ultimate  resis- 
tances given  are  the  means  of  seven  sets  of  tests,  eight  in 
each  set.  Half  bricks  were  tested  flatwise.  This  mode  of 
testing  obviously  yields  much  higher  values  than  if  the 
bricks  were  tested  on  edge. 


] 

>bs.  per  sq.  in 

Max. 

Mean. 

Min. 

Common  Hudson  River,  moulded  
Stiff  Clay,  side  cut  

4,357 

3,203 

2,006 
2,072 

Harvard  over-burned  

' 

6,642 

The  hollow  tiles  were  of  two  types,  six-core  and  two-core. 
The  cross-sections  were  10  inches  by  12  inches,  8  inches  by 
12  inches,  8  inches  by  16  inches,  and  12  inches  by  12  inches. 
The  length  or  height  of  each  set  of  tiles  was  12  inches  with 
one  exception  of  8  inches.  The  tiles  were  all  tested  with 
the  webs  (or  cores)  vertical  and  the  net  sectional  areas 


Art.  68.]  BRICKS  AND  BRICK  PIERS.  413 

varied  from  about  41  square  inches  to  60  square  inches. 
The  ultimate  resistances  per  square  inch  on  both  the  net 
sections  and  the  gross  sections  are  as  given  below.  There 
were  five  sets  of  ten  tests  each  and  the  results  given  are 
the  greatest,  mean  and  least  results  of  the  five  sets. 


- 

vbs.  per  sq.  in 

Max. 

Mean. 

Min. 

Net  section 

•  S  7i8 

4.  S08 

3  826 

Gross  section 

2,680 

2,090 

i  710 

Brick  Piers. 

Inasmuch  as  tests  of  brick  piers  have  shown  that 
their  ultimate  compressive  resistances  run  only  from  about 
1000  to  4500  pounds  per  square  inch,  depending  upon 
the  character  of  the  mortar,  it  is  seen  that  in  such  masonry 
a  small  portion  only  of  the  compressive  resistance  of  the 
bricks  is  developed  in  piers  and  other  similar  brick-masonry 
masses. 

These  latter  results  doubtless  depend  largely  upon  the 
cementing  material.  There  is  no  question  that  the  ulti- 
mate resisting  capacity  of  brick  masonry  is  affected 
greatly  by  the  resisting  capacity  of  the  mortar,  and 
the  same  general  observation  can  be  applied  to  other 
classes  of  masonry.  There  is  more  than  this,  however, 
affecting  the  carrying  capacity  of  brick  and  other  grades 
of  masonry  as  compared  with  the  ultimate  compressive 
resistance  of  the  bricks  used  in  the  one  case  of  masonry 
or  of  the  individual  stones  employed  in  the  other.  The 
texture  or  character  of  the  mass  of  burned  clay  com- 
posing the  brick  is  exceedingly  variable,  both  in  conse- 
quence of  the  varying  mixture  of  the  material  in  the  bricks 


414  COMPRESSION.  [Ch.  VIII. 

before  being  burned  and  in  consequence  of  the  varying 
degree  of  burning  in  each  individual  brick.  Again,  what- 
ever may  be  the  care  in  placing  the  bricks  in  a  testing- 
machine,  including  the  cushioning  of  the  ends,  it  is  prac- 
ticably impossible  to  secure  anything  like  a  uniform  bear- 
ing upon  either  the  ends,  sides,  or  beds.  Their  irregular 
dimensions  and  exterior  surfaces  and  the  varying  quality 
of  the  materials,  even  in  the  best  of  brick,-  introduce  into 
their  resisting  capacity  elements  of  variation  which  are 
frequently  so  great  as  to  lead  to  abnormal  results.  While 
the  mortar  used  in  forming  a  mass  of  brick  masonry  un- 
doubtedly fills  up  many  irregularities  of  surface,  voids 
of  considerable  magnitude  frequently  remain  unfilled. 
The  consequence  of  these  uncontrollable  elements  in  a 
mass  of  brick  masonry  is  always  a  'material  reduction  of 
ultimate  carrying  capacity  and  frequently  a  large  reduc- 
tion. However  excellent  in  quality,  therefore,  the  mor- 
tar or  binding  materia1  in  a  brick-masonry  pier  may  be, 
it  is  inevitable  that  there  will  be  not  only  a  wide  range  in 
ultimate  compressive  resistance,  but  in  all  cases  a  material 
reduction  below  that  exhibited  by  the  individual  bricks 
when  tested  by  themselves. 

Profs.  Arthur  N.  Talbot  and  Duff  A.  Abrams  reported, 
in  Bulletin  No.  27  (1908)  of  the  University  of  Illinois,  the 
results  of  a  series  of  sixteen  tests  of  brick  piers  and  the 
same  number  of  hollow  terra-cot ta  block  piers.  Two  grades 
of  brick  were  used,  a  hard-burned  shale  brick  and  a  soft 
under-burned  clay  brick.  Eighteen  of  the  former  tested 
on  beds  gave : 


] 

^bs.  per  sq.  in 

Max. 

Mean. 

Min. 

Ult  Comp.  Resist  

14,1  50 

10.690 

7,01,0 

Art.  68.]  BRICKS  AND  BRICK  PIERS. 

Sixteen  of  the  soft  bricks  similarly  tested  gave: 


] 

^bs.  per  sq.  in 

Max. 

Mean. 

Min. 

Ult  Comp  Resist       

5,670 

T.Q2O 

2,190 

The  hollow  terra-cotta  blocks  were  about  4  inches  by 
8  inches,  4  inches  by  8|  inches  and  4  inches  by  8|  inches 
in  cross-section,  the  height  or  length  being  generally  8  inches, 
but  4  inches  in  some  cases.  These  blocks  had  three  cores, 
two  1 1  inches  square  each  and  one  i^  inches  by  £  inch. 

TABLE    I 
AVERAGE   VALUES   FOR   BRICK   COLUMNS 


Columns. 

Average 
Ultimate. 
Load,  Ib. 
per  sq.  in. 

Ratio  of 

Ultimate 
of  Col- 
umn to 
Ultimate 
of  Brick. 

Ratio  of 
Ultimate 
of  Col- 
umn to 
Ultimate 
of  "A" 

E 
Initial 
Modulus 
of 
Elasticity. 

Num- 
ber of 
Tests. 

Shale  Building  Brick. 


A-Well  laid,  1-3  portland 

cement  mortar,  67  days 

3365 

•31 

I.  00 

4,780,000 

3 

Well  laid,  1-3  portland  ce- 

ment mortar,  6  months. 

3950 

•37 

1.18 

5,025,000 

2 

Well  laid,  1-3  portland  ce- 

ment     mortar,     eccen- 

trically loaded,  68  days. 

2800 

.26 

•83 

4,4OO,OOO 

2 

Poorly  laid,  1-3  portland 

cement  mortar,  67  days 

2920 

.27 

.87 

3,525,000 

2 

Welllaid,  1-5  portland  ce- 

ment mortar,  65  days.  . 

2225 

.21 

.66 

3,250,000 

2 

Well  laid,  1-3  natural  ce- 

ment mortar,  67  days.  . 

1750 

.16 

•52 

8OO,OOO 

I 

Well  laid,  1-2  lime  mortar, 

66  days 

I  A  CQ 

14. 

AT. 

IO4,OOO 

2 

T"O 

.    Xif 

•  T-«J 

Under-burned  Clay  Brick. 


Well  laid,  1-3  portland  ce- 
ment mortar,  63  days.  . 

1060 

.27 

•3i 

433>ooo 

2 

4i6 


COMPRESSION. 


[Ch.  VIII. 


The  brick  columns  were  about  12 J  inches  by  12^  inches 
in  section  and  10  feet  long.  The  mortar  joints  were  about 
f  inches  thick.  Failure  of  these  columns  took  place  chiefly 
by  vertical  cracks  through  joints  and  bricks.  Table  I  gives 
the  mean  results  of  these  tests. 

The  characteristics  and  dimensions  of  the  terra-cotta 
columns  or  piers  and  the  average  results  of  tests  per  square 
inch  of  gross  area  are  given  in  Table  la. 

TABLE  IA. 
AVERAGE  VALUES  FOR  TERRA  COTTA  COLUMNS 


Ratio 

Average 

Ultimate 

E 

Characteristics 

Number 
of 

Ultimate 
Unit 

Column 

Initial 

of  columns. 

Columns 
in  Average 

Load 
Ib.  per 
sq.  in. 

Ultimate 
of  Block 
(Gross 

Modulus 
of 
Elasticity. 

area)  . 

1-2  portland  cement  mortar.     All  well  laid  and  centrally  loaded. 


§1X8^  in 

2 

288s 

8l 

2  ,  1  94.  OOO 

85X13  in 

2 

^070 

80 

2  ,  1  94  OOO 

T  -2  v  T  -j  in 

2 

29SS 

.8<5 

2,I94,OOO 

125  X  123  in.,    1-3  portland  cement  mortar,  well  laid  unless  noted. 


Central  load          

2 

^79O 

•  74 

2,  76s,  ooo 

* 

4.^00 

8^* 

Eccentric  load. 

I 

^4.70 

.65 

2,^0,000 

Poorly  laid,  central  load  
Poorly  laid  eccentric  load        .... 

i 
i 

3305 

^IIO 

.64 
.60 

3,200,000 
2,500,00.0 

Inferior  blocks,  central  load  
I—  e  mortar,  central  load  

i 

2 

3050 

335° 

•59 
.65 

2,300,000 
2,690,000 

*  Estimated. 


The  average  age  of  columns  when  tested  was  67  days. 

The  joints  of  the  columns  were  about  f  inch  thick  and 
the  blocks  were  laid  on  end.  Failures  were  sudden  and 
accompanied  or  caused  by  longitudinal  cracks.  In  fact, 


An  8  X  i6-in.-face  brick  pier  with  i6-in.  square  base 
laid  in  lime  mortar.  It  was  tested  at  the  U.  S. 
Arsenal,  Watertown,  Mass.,  and  gave  an  ultimate 
compressive  resistance  of  1233  Ibs.  per  sq.  in.  on 
the  upper  section  and  601  Ibs.  per  sq.  in.  on  the 
lower  section.  The  cracks  due  to  failure  are  clearly 
seen. 

(To  face  page  417.) 


Art.  68.1 


BRICKS  AND  BRICK  PIERS. 


417 


the  chief  manner  of  fracture  of  both  brick  and  terra-cotta 
columns  or  piers  is  by  longitudinal  cracking. 

Table  II  exhibits  the  results  of  testing  piers  of  brick 
masonry  in  the  Gov't  testing  machine  at  Watertown,  Mass. 
It  is  taken  from  "  Ex.  Doc.  No.  35,  4pth  Congress,  ist 
Session."  The  dimensions  of  piers  are  shown  in  the  table; 
also  the  kinds  of  mortar  used  and  the  grades  of  brick. 
The  "  common  "  and  "  face  "  brick,  both  hard  burnt, 
were  from  North  Cambridge,  Mass.  The  other  bricks 

TABLE    II. 
CRUvSHING  STRENGTH  OF  BRICK  PIERS. 


No. 

Height 
of  Pier, 
Ft.   Ins. 

Section 
of  Pier, 
Ins. 

Composition  of  Mortar. 

Weight 
per 
Cu.  Ft., 
Lbs. 

Ultimate 
Resistance, 
Lbs.  per 
Sq.  In. 

i 

i         4 

8X8 

lime,  3  sand. 

137-4 

,520! 

2 

6        8 

8X8 

'        3 

1  13  •  5 

877 

3 

i        4 

8X8 

Portland  cement,  3  sand. 

J«3  •  3 

136.3 

,776 

«5 

.4 
5 

6        8 

2           0 

8X8 
12X12 

lime,  3  sand. 

133-5 

,249 
,940 

1 

6 

2           0 

12X12 

3      ' 

,900 

rQ 

7 

10           0 

12X12 

.','       3      «'«' 

131  .7 

O 

8 
9 

10        o 

2           O 

I2X  12 
I2XI2 

Portland  cement,  2  sand. 

125  .0 

,'807 
,670 

to 

10 

10           0 

I2XI2 

2         " 

132.2 

*    ' 
,253J 

ii 

i        4 

8X8 

lime,  3  sand. 

135.6 

4   o) 

12 

6        8 

8X8 

3      ' 

133.6 

,540 

ja 

13 

2           0 

I2X  12 

3      ' 

y 

14 

2           0 

I2X  12 

3      ' 

,050 

_O 

15 

9        9 

I2XI2 

"       3      " 

131.5 

,118  !•  2 

16 
17 
18 

10           0 
10           0 
2           8 

I2XI2 
I2X  12 

i6X  16 

Portland  cement,  2  sand. 

2        ' 

136.0 
131  .0 

,587 
,003 

,7  20 

1 

19 

10           0 

16X16 

2        ' 

0 

20 
21 
22 

2           O 

6        o 
6        o 

12X12 
12X12 
12X12 

lime,  3  sand. 
3      ' 

3      ' 

119.7 

,370)     Bay 
,133  >•  State 
,210)  bricks. 

23* 

6        o 

12X12 

lime,  3  sand. 

118.  2 

33il 

24t 

6        o 

12X12 

3      ' 

118.1 

,211 

25 

7      10 

12X12 

3      ' 

1  20  .  3 

(i  74 

26 

10           0 

12X12 

3      ' 

118.0 

924 

J2 

27 

10           0 

8X12 

3      ' 

107  .0 

940 

•rj 

28 
29 

30 

TO            O 

6        o 
6        o 

12X16 
12X12 
12X12 

3      ' 
3     "    .  i  Rosendale  cement. 
Rosendale  cement,  2  sand. 

118.7 

120.6 

123.0 

773 
,646 
»97  2 

rl 

0) 

31 
32 

33 

6        o 
6        o 
6        o 

I2X  12 

I2XI2 
I2X  12 

lime,  3  sand,  2  Portland  cement. 
Portland  cement,  2  sand. 
Clear  Portland  cement- 

120.3 
119.7 
126.6 

,411 
,792 

,375 

fe 

*  Joints  broken  every  6  courses. 


t  Bricks  laid  on  edge. 


41 8  COMPRESSION.  [Ch.  VIII. 

were  from  the  Bay  State  Brick  Co.,  of  Boston  and  Cam- 
bridge, Mass.,  and  were  medium  burnt. 

The  brick  piers  were  built  of  bricks  "laid  on  beds  and 
joints  broken  every  course,  with  the  exception  of  two  12  by 
12  piers,  one  of  which  had  joints  broken  every  sixth  course, 
and  one  had  bricks  laid  on  edge. 

"They  were  built  in  the  month  of  May,  1882,"  and 
"their  ages  when  tested  ranged  from  14  to  24  months." 
They  were  all  tested  between  cast-iron  plates. 

"  Loads  were  gradually  applied  in  regular  increments, 
.  .  .  returning  at  regular  intervals  to  the  initial  load.  .  .  . 
Cracks  made  their  appearance  at  the  surfaces  of  the 
piers  and  were  gradually  enlarged  before  the  maximum 
loads  were  reached.  Final  failure  occurred  by  the  partial 
crushing  of  some  of  the  bricks,  and  by  the  enlargement  of 
these  cracks,  which  took  a  longitudinal  direction  and 
occurred  in  the  bricks  of  one  course  opposite  the  end  joints 
of  the  bricks  in  the  adjacent  courses.  This  manner  of 
failure  was  common  to  all  piers. 

It  is  important  to  notice  that  the  resistance  of  the  piers 
varies  with  the  strength  of  the  mortar  used  in  the  joints. 

Brick  piers,  8  inches  by  8  inches  in  cross-section  and 
6  feet  high,  built  of  Hudson  River  common  brick,  and 
others  of  Sykesville  face  brick  were  tested  to  destruction  in 
the  testing  laboratory  of  the  Department  of  Civil  Engineer- 
ing of  Columbia  University  in  1915  by  Mr.  J.  S.  Macgregor, 
in  charge  of  the  laboratory,  with  the  following  results,  two 
of  the  piers  being  built  of  Hudson  River  common  brick  and 
three  of  the  Sykesville  face  brick. 


Lbs.  per  sq.  in 

. 

Max. 

Mean. 

Min. 

Hudson  River  Common  
Sykesville  Face  

902 
3,436 

812 
3,363 

722 
3,289 

Art.  68.]  BRICKS  AND  BRICK  PIERS.  419 

These  piers  also  gave  the  two  following  values  for  the 
modulus  of  elasticity  in  compression : 

Hudson  River  Common E=    748,000  Ibs.  per  sq.  in. 

Sykesville  Face £  =  2,860,000  Ibs.  per  sq.  in. 

The  age  of  the  columns  was  60  days.  The  ends  were 
finished  with  plaster  of  paris  to  secure  square  and  uniform 
bearings.  The  two  moduli  were  determined  at  intensities 
of  stress  less  than  250  pounds  per  square  inch. 

Mr.  Macgregor  also  obtained  the  ultimate  resistances  of 
three  piers,  74  inches  high  built  up  of  single,  approximately 
8 -inch  by  12 -inch  hollow  tile  giving  a  gross  horizontal  cross- 
section  of  94  square  inches  and  a  net  section  of  actual  tile 
material  of  50  square  inches. 

These  tile  piers  had  f-inch  joints  filled  with  Portland 
cement  mortar,  i  cement,  3  sand,  the  age  of  the  piers 
being  60  days. 

The  ultimate  compressive  resistances  per  square  inch 
for  the  three  piers  were  as  follows : 

Gross  Section 1,236;  1,239;  and  1,117  Ibs.  per  sq.  in. 

Net  Section 2,324;   2,329;  and  2,100  Ibs.  per  sq.  in. 

These  tile  piers  failed  in  the  blocks  in  most  cases,  but 
in  other  cases  in  the  joints.  The  failures  of  the  blocks 
showed  vertical  cracks  as  well  as  horizontal  and  some  spalling. 

The  results  of  all  the  experimental  investigations 
available  in  connection  with  brick  masonry  and  experiences 
in  the  best  class  of  engineering  work  indicate  that  masonry 
laid  up  of  good  hard-burnt  common  brick  may  safely 
carry  a  working  load  of  15  to  20  tons  per  square  foot  or 
210  to  280  pounds  per  square  inch.  In  the  construction 
of  this  class  of  masonry  where  the  duties  are  to  be  severe  it 
is  of  the  utmost  importance  that  the  best  class  of  Portland 
cement  mortar  be  employed,  as  the  carrying  capacity  of 
brick  masonry  depends  largely  if  not  chiefly  upon  the 
character  of  the  mortar. 


420  COMPRESSION.  [Ch.  VIII. 

Art.  69. — Natural  Building  Stones. 

The  ultimate  compressive  resistance  of  natural  building 
stones  is  affected  greatly  by  the  condition  of  the  rock 
from  which  the  cube  or  other  test -piece  is  taken.  That 
portion  of  a  ledge  exposed  to  the  weather  may  be  much 
weakened  and,  in  fact,  even  disintegrated,  but  the  material 
at  a  short  distance  from  the  exterior  surface  may  have  the 
greatest  resistance  of  which  the  particular  kind  of  stone  is 
capable  of  yielding.  Again,  the  compressive  resistance 
of  stones  on  their  natural  beds  is  much  greater  than  when 
tested  on  edge.  In  the  tests  which  follow  the  test-pieces 
were  fairly  representative  of  such  quality  of  stones  as 
would  pass  inspection  in  first-class  engineering  work,  and 
it  is  to  be  assumed  that  they  were  compressed  on  their 
beds  unless  otherwise  stated. 

Table  I  taken  from  the  "  U.  S.  Report  of  Tests  of  Metals 
and  Other  Materials"  for  1894,  exhibits  the  coefficients  of 
elasticity,  ultimate  compressive  resistances,  weights  per 
cubic  foot  and  coefficients  of  thermal  expansion  per  degree 
Fahr.,  as  well  as  the  ratio,  r,  between  lateral  and  direct 
strains  for  the  granites,  marbles,  limestones,  sandstones, 
and  other  stones  shown  in  the  left-hand  column.  The 
coefficients  of  elasticity  and  of  thermal  expansion  were 
determined  by  employing  blocks  of  stone  about  24  ins. 
long  and  6  ins.  by  4  ins.  in  cross-section,  the  gauged  length 
being  20  inches,  but  the  ultimate  compressive  resistances 
were  found  by  testing  4-inch,  cubes.  The  number  of  tests 
for  each  coefficient  of  elasticity  and  ultimate  resistance 
varied  from  one  to  nine  but  were  generally  two  or  three. 
The  general  run  of  values  of  ultimate  resistance  will  be 
found  to  conform  as  well  as  could  be  expected  with  results 
for  the  same  kind  of  stones  in  the  tables  which  follow. 


3 


.S  " 


a;  •_£ 


11 


NATURAL  BUILDING  STONES. 


421 


Art.  69.] 

It  will  be  observed  that  the  marbles  are  the  heaviest  stones, 
although  the  granites  are  not  much  lighter.  There  is  a 
large  difference,  however,  between  the  sandstones  and  the 
marbles  or  granites. 

TABLE  I. 

NATURAL  STONES  IN  COMPRESSION  ON  BEDS. 


Stone. 

Coefficient 
of 
Elasticity, 

Ultimate  Compressive 
Resistance,  Lbs.  per 
Sq.  In. 

Weight 
per 
Cu.  Ft., 

Coefficient 
of  Expan- 
-    sion  per 

r. 

Lbs.  per 
Sq.  In. 

Max. 

Mean. 

Min. 

Lbs. 

Fahr. 

Branford  granite,  Conn  .... 

8,712,100 

15,854 

15,707 

i5,56o 

'162 

.00000398 

i 

4 

Milford  granite,  Mass  

7,676,750 

25,738 

23,773 

19,258 

162.  5 

.00000418 

I 

sTs 

Trov  granite    N    H    .  .  . 

6  118  850 

28  768 

26  i  74 

23  580 

164  .  7 

.  00000337 

1  . 

Milford  pink  granite,  Mass.  . 

6,200,350 

22,162 

18,988 

15,756 

161  .9 

5-1 

Creole  marble,  Ga  

7,993,7°° 

15,512 

1  3,466 

1  1  ,420 

1  70 

r 

2^9 

Cherokee  marble,  Ga  

10  427,800 

13,415 

12,619 

11,822 

167.8 

.00000441 

3-7 

Etowah  marble,  Ga  

8,792,600 

14,217 

14,053 

13,888 

169.8 



i 
3^6 

Kennesaw  marble,  Ga  

8,217,950 

10,771 

9,563 

8,354 

168.1 

i 

T                          "U1          T\f 

•j   y 

Marble  Hill  marble,  Ga.  .  .  . 

9,950,850 

,,53, 

n,505 

11,478 

1  68  .  6 

.00000202 

i 
3-4 

Tuckhoe  marble,  N.  Y  

15  173,200 

19.223 

16,203 

1  1  ,640 

178 

.  OOOOO44I 

4-5 

Mount  Vernon  limestone,  Kv 

3,278,400 

11,566 

7,647 

5,247 

139-  1 

.00000464 

L 

4 

Oolitic  limestone,  Ind  



_  



__ 



.00000437 

North  River  bluestone,  N.  Y. 

5,475,300 



22,947 

.  —  .  



.00000519 

Manson  slate,  Maine  







. 



Cooper  sandstone,  Oregon  .  . 

3,021,350 

16,366 

15,284 

14,203 

159-8 

.  oooooi  77 

i 
ii 

Maynard  sandstone,  Mass..  . 

2,034,650 

10,538 

9,880 

9,223 

133-5 

.00000567 

3 

Kibbe  sandstone,  Mass  

2,066,800 

10,663 

10,363 

10,063 

133-4 

.00000577 

i 
3-3 

Worcester  sandstone,  Mass.  . 

2,668,750 

9,869 

9,763 

9,656 

136.6 

.00000517 

i 

4-4 

Potomac  sandstone,  Md.  .  . 











.000005 

13,441 

I  0  ,  2  7  0 

*  Yammerthal      flint      lime- 

• 



28,647 

From  Report  of  1899. 


422  COMPRESSION.  [Ch.  VIII. 

The  coefficients  of  elasticity  generally  range  considerably 
higher  than  those  for  concrete  in  Art.  67,  but  the  sand- 
stones form  an  exception  to  this  observation.  The  coeffi- 
cients of  thermal  expansion  vary  between  rather  wide 
limits  but  they  are  mostly  a  little  lower  only  than  those 
determined  for  concrete.  The  coefficient  for  the  Dycker- 
hoff  cement  is  very  close  to  those  exhibited  for  cement 
mortar  and  concrete  in  Art.  60.  The  column  headed  r, 
giving  the  ratios  between  lateral  and  direct  strains,  contains 
interesting  data.  From  what  has  been  shown  in  Art.  4 
it  is  apparent  that  the  total  volume  of  the  test-pieces  was 
considerably  reduced  by  the  compression  to  which  the 
cubes  were  subjected. 

The  coefficients  of  elasticity  were  determined  at  in- 
tensities of  pressure  running  from  1000  or  2000  pounds 
per  square  inch  up  to  8000  or  10,000  pounds  per  square 
inch. 

A  coefficient  would  first  be  determined  at  comparatively 
low  pressures,  as  from  1000  to  3000  pounds  per  square 
inch,  and  then  at  higher  pressures,  as  from  7000  to  9000 
or  10,000  pounds  per  square  inch.  As  a  rule,  the  co- 
efficients determined  at  the  higher  pressures  were  mate- 
rially higher  in  value  than  the  others,  the  stiffness  of  the 
stone  increasing  with  the  loads  within  the  limits  of  the  test. 
The  values  in  the  table  are  the  means  of  those  at  the  low 
and  high  pressures. 

With  the  ordinary  working  values  of  pressures  in 
masonry,  probably  not  more  than  two  thirds  of  the 
values  of  the  coefficients  of  elasticity  given  in  the  table 
should  be  employed. 

In  the  "  U.  S.  Report  of  Tests  of  Metals  and  Other  Mate- 
rials "  for  1900  there  may  be  found  the  results  of  compress- 
ing 4-inch  cubes  of  Tennessee  marble  and  of  granite  from 
the  Mount  Waldo  Quarries  at  Frankfort,  Maine.  The 


Art.  69-]  NATURAL  BUILDING  STONES.  423 

ultimate  compressive  resistances  of  the  4-inch  Tennessee 
marble  cubes  expressed  in  pounds  per  square  inch,  were  as 
follows : 

Maximum.  Mean.  Minimum. 

25,478  20,329  16,309 

The  preceding  three  results  cover  twenty  tests. 

The  ultimate  resistances  in  pounds  per  square  inch 
of  the  "  Black  Granite"  from  the  Waldo  Quarries,  as 
determined  from  four  tests  of  2 -inch  cubes,  were  as  follows: 

Maximum.  Mean.  Minimum. 

32>635  3°>949  29,183 

Again,  in  the  same  report,  the  ultimate  resistances  in 
pounds  per  square  inch  of  four  4-inch  cubes  of  limestone 
from  Carthage,  Mo.,  are  as  follows* 

Maximum.  Mean.  Minimum. 

17,130  14,947  i3»66° 

The  preceding  tests  and  the  results  of  others  given  in 
Table  II  have  been  determined  by  compressing  cubes  4 
inches  and  5  inches  on  the  edge  and  it  has  been  generally 
customary  to  use  a  cube  for  a  test  piece  for  either  natural 
or  artificial  stones.  It  has  already  been  indicated,  however, 
in  Art.  62  that  such  a  short  test  piece  in  compression  must 
necessarily  give  higher  results  than  should  be  credited  to 
the  material. 

The  use  of  compressive  test  specimens  with  lengths 
two  to  two  and  one-half  times  the  diameter  is  just  begin- 
ning, but  that  use  has  not  become  sufficiently  general,  nor 
has  it  been  long  enough  the  practice,  to  make  available 
results  from  such  desirable  tests. 

Furthermore,  some  tests  have  shown  that  ultimate  com- 
pressive resistances  may  be  materially  higher  for  large  cubes 


424 


COMPRESSION. 


[Ch.  VIII. 


than  for  small  ones.  This  is  probably  due  to  the  lateral 
supporting  effect  given  to  parts  of  the  test  piece  by  the 
friction  between  the  bearing  head  of  the  machine  and  the 
face  of  the  material  under  "test  with  which  it  is  in  contact. 
Preferably  no  cube  tested  for  engineering  purposes  should 
be  less  than  12  by  12  inches  in  section,  nor  should  any  test 
piece  be  shorter  than  twice  its  diameter. 

The  results  found  in  Table  II  are  taken  from  the  "  U.  S. 
Report  of  Tests  of  Metals  and  Other  Materials,"  for  1894. 
They  relate  to  the  various  kinds  of  rock  indicated  and 
were  found  by  testing  4-inch  to  5 -inch  cubes  on  their  beds. 

TABLE  II. 


State. 

Stone. 

Ultimate 
Compressive 
Resistance,- 
Pounds  per 
Square  Inch. 

Minnesota 

Ortonville  granite 

2O  4.1  ^ 

Kasota  pink  limestone 

108^"} 

Faribault  marble 

17  ?8O 

Duluth  brownstone 

4T  C  T 

Mankato  sandstone             .    .        

9  606 

Mantorville  sandstone             

8  77S 

Frontinac  sandstone            

10  1  14 

Luverne  quartzite        

21   556 

IQ.87S 

Iowa 

Rubble  rock 

94.6  ^ 

Firestone 

4-8^4. 

Gypsum  Fort  Dodge 

2  8OQ 

The  ultimate  resistances  of  the  sandstones  are  relatively 
low,  while  the  higher  values  are  found  for  granites,  lime- 
stones, and  quartzites,  as  is  usual. 

In  1906  the  Carnegie  Institution  of  Washington  pub- 
lished An  Investigation  into  the  Elastic  Constants  of  Rocks, 
More  Especially  with  Reference  to  Cubic  Compressibility, 
by  Mr.  Frank  D.  Adams  and  Dr.  Ernest  G.  Coker.  The 
experimental  part  of  this  investigation  was  made  atMcGill 
University  under  the  auspices  of  the  Carnegie  Institution. 


Art.  69.] 


NATURAL    BUILDING  STONES. 


425 


Although  this  investigation  was  made  as  a  contribution 
more  to  physics  than  to  engineering,  the  results  obtained 
are  of  both  interest  and  value  to  engineers  and  it  is  well 
to  make  use  even  for  engineering  purposes  of  results  deter- 
mined with  so  much  care  and  such  extreme  accuracy  in 
spite  of  the  fact  that  the  specimens  used  were  only  i  inch 
square  in  section  or  i  inch  in  diameter  and  3  inches  long. 
If  E  is  the  ordinary  modulus  of  elasticity  in  compression 
G  the  modulus  of  elasticity  for  shearing,  V  the  so-called 
bulk  modulus,  i.e.,  the  reciprocal  of  the  rate  of  change  of 
unit  volume  for  unit  intensity  of  stress,  and  r  the  ratio  of 
the  rate  of  lateral  strain  of .  the  specimen  divided  by  the 
rate  of  direct  strain  under  compression,  Table  III  gives 
the  results  of  these  experimental  determinations  for  those 
materials  which  American  engineers  more  commonly  use. 


TABLE    III. 


Specimen. 

E. 

^ 

G. 

v        E' 

3(1  -2r) 

Black  Belgian  marble  . 

11,070,000 

0.2780 

4,330,000 

8,303,000 

Carrara  marble  

8,046,000 

0.2744 

3,154,000 

5,946,000 

Vermont  marble  

7,592,000 

o  .  2630 

3,000,000 

5,341,000 

Tennessee  marble  

9,006,000 

0.2513 

3,607,000 

5,967,000 

Montreal  limestone.  .  . 

9,205,000 

0.2522 

3,636,000 

6,167,500 

Baveno  granite 

6,833,000 

O2  ^28 

2.724,800 

4  604  ooo 

Peterhead  granite  .... 

8,295,000 

•  ^  O     ^ 

0.2112 

3,399,ooo 

4,792,000 

Lily  Lake  granite  

8,165,000 

o.  1982 

3,380,000 

4.517,500 

Westerly  granite  
Quincy  granite  (i)..  .  . 

7,394,500 
6,747,000 

0.2T95 
0.2152 

3.019,700 
2,781,600 

4,397.500 
3,984,000 

Quincy  granite  (2).  .  .  . 

8;247,5oo 

0.1977 

3,445  ooo 

4.555,000 

Stanstead  granite  

5,685,000 

0.2585 

2,258,700 

3,940,000 

Ohio  sandstone  

2,290,000 

o  .  2900 

888,000 

1.816,000 

Plate  glass  

10,500.000 

O   227^ 

4  290  ooo 

6  448  ooo 

w  •  '•*  /  o 

426  COMPRESSION.  [Ch.  VIII. 

•^ 

Art.  70. — Timber. 

The  ultimate  compressive  resistance,  coefficient  of  elas- 
ticity, and   other   physical   properties   of  timber   in   com- 
pression are  affected  greatly  by  the  amount  of  moisture 
in  the  timber  and  by  the  size  of  stick.     The  investigations 
of  Professor  J.  B.  Johnson,  acting  for  the  Forestry  Division 
of  the  U.  S.  Department  of  Agriculture,  have  shown  that 
when   the    amount   of   moisture   exceeds   about    30%    by 
weight  of  the  timber  the  physical  properties  are  not  much 
affected  by  any  increased    saturation.     The  walls  of  the 
wood  cells  at  that  point  seem  to  experience  their  maximum 
softening.      Green  timber  may  be  considered  as  carrying 
about  one  third  of  its  weight  in  moisture,  and  it  seems  to 
matter  little  whether  that  moisture  is  water  or  sap,  timber 
once  dried  and  resaturated  appearing  to  suffer  the  same 
diminished  resistance  as  in  its  original  green  condition. 
Professor  Johnson's  tests  showed  that  the  Southern  pines 
increased  their  ultimate   compressive  resistance  in   some 
cases  as  much  as  75%  by  the  process  of  drying  or  seasoning 
from  33%  of  moisture  down  to  10%,  the  general  rule  being 
a  greatly  increased  compressive  resistance  with  a  decrease  of 
moisture.     It  follows  from  these  results,  therefore,  that  green 
timber  will  be  much  weaker  in  compression  than  seasoned 
timber.     Ordinary  air  seasoning  even  under  cover  seldom 
reduces  moisture  below  about  15%  in  weight  of  the  timber 
itself,  although  under  favorable  circumstances  of  seasoning 
the  moisture  may  sometimes  drop  to  12%  of  that  weight. 
As  a  matter  of  precision,  therefore,  or   accuracy,  the  ulti- 
mate   compressive    resistance    of    timber    should    always 
be  stated  in  connection  with  the  percentage  of  moisture 
carried  by  the  timber.     This  will  be  found  to  be  the  case 
in  all  of  Professor  Johnson's  experimental  work,  to  which 
reference  has  already  been  made  and  the  results  of  which 


Art.  70.]  TIMBER.  427 

are  chiefly  found  in  bulletins  Nos.  8  and  15  of  the  Division 
of  Forestry  of  the  U.  S.  Department  of  Agriculture,  the 
former  being  dated  1893. 

The  earlier  tests  of  Professor  Johnson  were  made  on  a 
basis  of  15%  moisture,  but  in  his  later  work  a  basis  of  12% 
moisture  was  adopted,  and  he  states  in  Circular  No.  15 
that  in  reducing  the  moisture  from  15%  to  12%  the  corre- 
sponding increases  in  the  ultimate  compressive  resistance 
in  pounds  per  square  inch  of  Southern  pines  are  approxi- 
mately as  follows: 


Endwise. 

Across  Grain. 

Long-leaf  pine 

I    IOO 

1  80 

Cuban  pine 

800 

2  2O 

Loblolly  pine                                             .  . 

QOO 

I  SO 

Short-leaf  pine                                           .    . 

6OO 

60 

While  it  is  important  as  a  matter  of  physics  to  recognize 
clearly  the  effect  of  moisture  upon  the  compressive  re- 
sistance of  timber,  it  is  of  equal  importance,  and  possibly 
of  greater  importance,  to  recognize  the  fact  that  in  engineer- 
ing practice,  except  in  specially  protected  cases,  the  timber 
used  in  structures  is  more  or  less  exposed  and  can  seldom 
or  never  be  depended  upon  to  contain  even  as  little  as  15% 
of  moisture,  and  with  some  conditions  of  weather  and  at 
some  seasons  of  the  year  it  may  contain  considerably  more. 
It  follows,  also,  that  the  condition  of  timber  as  to  moisture 
in  most  structures  will  change  materially  from  time  to  time. 
It  would  be  unwise,  therefore,  and  perhaps  dangerous  to  use 
working  compressive  resistances  based  upon  the  results  of 
tests  of  small  pieces  with  moisture  reduced  to  15%  or  12%. 

Again,  it  has  been  frequently  stated  as  a  result  of  the 
timber  investigations  by  the  Forestry  Division  of  the 
U.  S.  Department  of  Agriculture,  that  the  ultimate  com- 


428  COMPRESSION.  [Chi  VIII. 

pressive  resistance  of  large  sticks  may  be  taken  as  practically 
identical  with  that  belonging  to  small  selected  test  pieces, 
the  quality  of  the  material  being  the  same  in  both  cases. 
It  is  possible,  if  the  quality  of  material  throughout  all 
portions  of  every  large  stick  were  identical  with  the  quality 
of  small  selected  specimens,  that  the  ultimate  compressive 
resistance  per  square  inch  might  be  the  same;  but  that  is 
radically  different  from  the  facts  as  they  are.  There  is 
probably  no  stick  of  timber  whose  condition  is  permanent 
at  any  given  time.  If  it  is  seasoning,  its  quality  is  im- 
proving, but  after  reaching  a  maximum  of  excellence  it 
begins  to  depreciate  by  decay  or  from  other  causes.  Any 
large  stick  of  timber  as  used  by  the  engineer  is  seldom 
free  from  some  point  of  incipient  decay  and  it  is  never 
free  from  knots,  large  or  small,  wind  shakes,  cracks  from 
one  cause  or  another,  or  from  some  other  defective  con- 
dition, at  some  point.  Small  specimens  for  testing  are 
invariably  so  selected  as  to  eliminate  such  spots  as  militating 
against  a  comparatively  high  resistance.  The  inevitable 
result  for  full-size  sticks  is  a  decreased  resistance  materially 
below  that  of  the  small  specimen.  For  all  these  reasons, 
therefore,  in  engineering  practice  it  would  be  a  radical 
error  to  accept  the  ultimate  compressive  resistance  per 
square  inch  of  small  test  specimens  as  practically  identical 
with  that  of  large  sticks.  Values  for  the  latter  class  of 
timber  should.be  determined  upon  pieces  as  large  as  those 
used  in  structures  and  under  the  same  conditions  in  which 
they  are  used,  which  means  an  indefinite  amount  of  moisture 
ordinarily  sensibly  larger  than  12%  or  15%. 

In  the  "U.  S.  Report  of  Tests  of  Metals  and  Other 
Materials"  for  1896  and  1897  there  may  be  found  results 
of  compressive  tests  for  coefficients  of  elasticity  for  sticks  of 
timber  as  shown  in  Table  I.  Those  sticks  were  many  of 
them  large  enough  to  form  full-size  posts.  They  appear  to 


The  fracture  of  a  piece  of  Douglass  fir  or  Oregon  pine  loaded  tangentially  to 
the  rings  of  growth.  The  ultimate  compressive  resistance  was  found  to  be  600 
Ibs.  psr  sq.  in. 

(To  face  page  429.) 


Art.  70.] 


TIMBER. 


429 


TABLE  I. 

TIMBER  IN  COMPRESSION. 


Kind  of  Wood. 

Coefficient  of  Elasticity, 
Pounds  per  Square  Inch. 

No.  of  Tests. 

Remarks. 

Maximum. 

Mean. 

Minimum. 

Douglas  fir  : 
Endwise  
Tangentially  
Radially  

3,461,000 
II2,OOO 
2O7,OOO 

1,789,000 

1,890,000 
2,252,000 

1,655,000 
1,623,000 
2,300,000 

2,358,000 
74,600 
158,000 

1,554,000 

1,657,000 
2,175,000 

1,469,500 
1,531,000 
2,251,000 

1,915,000 
40,000 
134,300 

1,338,000 

1,488,000 
2,049,000 

I,2O2,OOO 
1,437,000 
2,207,000 

4 
9 
6 

6 

4 
4 

6 

10 
12 

Not  well  seasoned. 

From  tops  of  trees 
From  butts  of  trees 

Not  well  seasoned 
Very  dry. 

White  oak  : 
Endwise 

Long-leaf  pine  : 
Endwise 

Short-leaf  pine  :  * 
Endwise     ......    .  . 

Spruce  :  * 
Endwise  
Old  yellow-pine  posts  :* 
Endwise  

*  These  results  are  means  of  determinations  at  intensities  varying  from 
500  to  5,000  pounds  per  square  inch. 

have  been  of  merchantable  timber  of  about  such  quality  as  is 
used  in  first-class  engineering  works.  They  had  the  usual 
supply  of  knots  and  other  features  which,  while  not  material 
defects,  prevented  the  pieces  from  being  of  selected  quality. 
As  also  shown  in  the  table,  there  were  a  considerable 
number  of  tests  in  each  case.  ''Endwise"  compres- 
sion means  compression  parallel  to  the  fibres  of  the 
timber,  while  "Tangentially"  means  a  direction  tangent  to 
the  rings  of  growth.  That  compression  indicated  by 
"Radially"  was  in  a  radial  direction,  i.e.,  passing  through 
the  centre  of  the  tree  trunk.  The  determinations  were 
made  at  intensities  of  pressure  varying  from  one  third  to 
one  half  the  ultimate  resistance.  It  will  be  noticed  that 
in  the  values  for  long-leaf  pine  the  highest  results  belong 
to  sticks  from  the  butts  of  trees,  while  those  from  the  tops 


430  COMPRESSION.  [Ch.  VIII. 

give  materially  less  values.  It  will  also  be  observed  that 
the  values  for  the  very  dry  yellow-pine  posts  in  the  last 
line  of  the  table  are  high,  showing  the  increased  stiffness 
due  to  the  absence  of  moisture.  The  coefficients  of  elas- 
ticity in  the  last  five  lines  of  the  table  were  computed 
from  the  resilience  of  the  compressed  columns  by  means 
of  a  formula  similar  to  eq.  (2)  of  Art.  44. 

The  values  of  the  elastic  limit,  ultimate  resistance  and 
modulus  of  elasticity  in  compression  along  the  fibres  as 
well  as  the  elastic  limit  in  compression  across  the  fibres  of 
nine  of  the  prominent  structural  timbers  of  the  United 
States,  both  for  large  or  structural  sizes  and  small  speci- 
mens, as  shown  in  Table  II,  are  taken  from  Tests  of  Struc- 
tural Timbers,  Forest  Service-Bulletin  108,  U.  S.  Depart- 
ment of  Agriculture,  by  Messrs.  McGarvey  Cline  and  A.  L. 
Heim,  1912,  and  exhibit  some  of  the  latest  experimental 
investigations  in  the  elasticity  and  resistance  of  timber. 
The  large  or  structural  sizes  had  cross-sections  up  to  10 
inches  by  16  inches  and  the  small  sizes  down  to  2  inches 
by  2  inches.  The  resistances  parallel  to  the  fibres,  i.e.  on 
end,  were  determined  for  pieces  whose  lengths  were  three 
to  four  times  the  cross  dimensions. 

The  authors  of  the  paper  properly  observe  that  the 
"  Results  of  tests  made  only  on  small  thoroughly  seasoned 
specimens  free  from  defects  "  "  may  be  from  one  and  one- 
half  to  two  times  as  high  as  stresses  developed  in  large 
timbers  and  joists."  This  is  an  important  conclusion  and 
a  number  of  results  in  Table  II  confirm  the  observations 
of  the  authors. 

It  is  essential  to  observe  the  small  resisting  capacity 
of  the  various  timbers  when  compressed  across  the  grain, 
the  resistance  in  the  latter  condition  being  but  a  small 
fraction  of  that  along  the  grain. 

Table  III  contains  the  results  of  tests  by  Colonel  Laidley, 


Art.  70.] 


TIMBER. 


431 


. 


10 

10 


r  Seasoned  Timber. 
Parallel  to  Grain. 
Pounds  per  Sq.  In. 


COOO 
OO 


« 


rt-  (N 
I-"   ON 


00   (N  O   O 

iO  O  COCO 

CL  °L  °.  *? 

rf  iO  v£~  \O~ 


00         rh 
CN   ON       HH 

^rf 


ON       OOO 
i-i        (NiO 


s 


o  o 

T^-O 

ooio 


o  :     o 

M."-1 


ts 


Green  Timber. 
arallel  to  Grai 
unds  per  Sq.  I 


t    r        O  CN       o     ' 

»-<  co        ^  (N         O 

MDt^      nn       o    ! 


88 


lOO  >OO  O^O  OO  OO  *OCN  (NO  iO^ 

g^co  cot^  MON  TT^  COON  lOON  0000  IOC 

^T  O^  rt*  iO  lO'O  O^CS  MI-I  cOcO  COON  lOiO 

cO^  COcO  COcO  CNcO  coco  COcO  coco  oToT 


o 

oo     . 
'     • 


O 
vO 


IOVO  O      .  O 

t^CN  iO.  O 

vq  O_  q     .  ^ 

cTco  ci  cT 


»OOO         rt-  O 

Oco        ONON 
ON  ON        i-    rj- 


10! 


o!oo!       HI]       o.' 
^    .       <"<.       co.       co. 


n  r'  nj 

cgl-         |-|    §  j;  .§        s 

^    W    W^    CO    10  ^    tO    w—  '    tfl    w    S-^W    55^    M'w^    S'MM    CO  'w    O.'  CO*  w 


43  2 


COMPRESSION. 

TABLE  III. 


[Ch.  VIII. 


No. 

Kind  of  Wood. 

Length, 
Inches. 

Compressed 
Section 
in 
Inches. 

Ultimate 
Resist- 
ance, 
Pounds 
per 
bquare 
Inch. 

Perpendicular 
to  or  with 
Grain. 

Remarks. 

16.5 

2-46X  2     0 

8  496 

With 

IQ  .  O 

I  .  21  X  I     21 

8253 

•  < 

4 
5 
6 

7 
8 

Oregon  maple  
Oregon  spruce  
California  laurel  
Ava  Mexicana  
Oregon  ash  

8.0 
24.02 
8.0 
8.0 
8.0 

3.63X3.63 
3.92X5.75 
3.58X3.58 
3.69X3.69 
3.64X3.64 

6,661 
5,772 
6,734 
6,382 

5,121 

« 

Unseasoned 
Worm-eaten 

9 

Mexican  white  mahogany  . 

8.0 
8  o 

3.77X3.77 

6,155 
4  814 

» 

8  o 

3.75X3   75 

« 

12 

13 

White  maple  
White  maple  
Red  birch  

12.0 
12.0 

13.0 

4.00X4.00 
4.00X4.00 
4.  26X4.  26 

7,140 
7,210 
8,030 

« 

Red  birch 

13   o 

4.  26  X  4   26 

7  820 

« 

16 

Whitewood  

I  2  .  O 

4.  oo  X  4  oo 

4  44° 

«« 

17 

18 

Whitewood  

12.0 

4.00X4-00 

4,330 

** 

I  2  .  O 

4  .  oo  X  4  oo 

5  760 

M 

20 

White  oak  
White  oak  

12.0 
12     O 

4.00X4.00 
4  .  oo  X  4  oo 

7,375 

,, 

22 

Ash  
Ash  .  .  . 

12.0 
I  2    O 

4-00X4-00 
4  .  oo  X  4  oo 

7,940 

7  640 

,, 

I  .  95 

3  •  45  X  3   oo 

i  ,i  50 

Perp. 

25 
26 
27 
28 

Oregon  maple  
Oregon  spruce  
Oregon  spruce  

3.63 
3-92 
3.92 
3.58 

3.63X3.00 
5.75X4.75 
4-75X4-00 
3-  58X  3   oo 

1,875 
710 
680 

Unseasoned 
Unseasoned 

29 

30 
3i 
32 
33 
34 

Ava  Mexicana  
Oregon  ash  
Mexican  white  mahogany  . 
Mexican  cedar  
Mexican  mahogany  
White  pine  

3-69 
3-64 

3-77 
3-75 
3-75 
3.06 

2  .  90 

3  -  69  X  3  •  oo 
3.64X3-00 
3-77X3-00 
3-75X3-00 
3  '75X3  -oo 
6-20X4-75 
4-75X4  oo 

2,100 
2,200 
2,150 
^950 
4,500 
875 

\ 

Mean  of  two 

36 
37 
38 
39 

Whitewood  
Whitewood  
Black  walnut  
Black  walnut  

3-  15 
3-15 
.875 
•875 
37!- 

4-75X6.20 
4-75X4-00 
4-75X4-00 
4.  oo  X  3.94 

900 
1,000 

2,450 

2,200 

\ 

Mean  of  two 
Mean  of  four 
Mean  of  two 
Mean  of  two 

4t 
42 

White  oak  

.40 

70 

4-  75  X  4.00 
4.75X4  oo 

3,550 

, 

Mean  of  four 

43 

Yellow  pine  •  - 

.90 

4.00X4.00 
4   05X4   oo 

I  ,9OO 

1 

i 

46 

Black  walnut  

•  25 

4.05  X4-oo 

.     2,400 

, 

48 

Black  walnut  

-75 
06 

4.05X4.00 

2,400 

, 

.  < 

5i 

White  pine  

.  oo 

4.05X4-00 

I  ,IOO 

, 

53 

White  pine  

< 

i 

55 

White  pine  

« 

56 

57 

Yellow  birch  
Yellow  birch  

4-25 

4.  25X3-00 
5    98  X  3   oo 

2,000 
I    650 

" 

58 
59 
60 

White  maple  
White  maple  
White  oak  

4.00 

4  .  oo 
3-95 

3.95X3-00 
5.98X3-00 
3.  96X3*-  oo 

1,700 
1,900 

2,500 

" 

Mean  of  two 

Art.  70.]  TIMBER.  433 

U.S.A.,  "Ex.  Doc.  No.  12,  47th  Congress,  2d  Session." 
A  few  other  tests  of  short  blocks  from  the  same  source  will 
be  found  in  the  t article  on  "Timber  Columns."  Unless 
otherwise  stated,  all  the  specimens  were  thoroughly  sea- 
soned. 

In  this  table  the  "length"  of  all  those  pieces  which 
were  compressed  in  a  direction  perpendicular  to  the  grain 
might,  with  greater  propriety,  be  called  the  thickness,  since 
it  is  measured  across  the  grain. 

In  the  tests  (24-60)  the  compressing  force  was  dis- 
tributed over  only  a  portion  of  the  face  of  the  block  on 
which  it  was  applied ;  thus  the  compressed  area  was  sup- 
ported, on  the  face  of  application,  by  material  about  it 
carrying  no  pressure.  In  some  cases  this  rectangular  com- 
pressed area  extended  across  the  block  in  one  direction, 
but  not  in  the  other.  In  all  such  instances  the  ultimate 
resistance  was  a  little  less  than  in  those  in  which  the  area 
of  compression  was  supported  on  all  its  sides. 

The  "ultimate  resistance"  was  taken  to  be  that  pressure 
which  caused  an  indentation  of  0.05  inch. 

Nos.  (44-5  5 )  show  the  effect  of  varying  thickness  of  blocks. 
Within  the  limits  of  the  experiments,  the  ultimate  resistance 
is  seen  to  decrease  somewhat  as  the  thickness  increases. 

The  best  series  of  values  of  the  ultimate  compressive  re- 
sistance of  timbers  as  actually  used  in  large  pieces  and  for 
engineering  structures  that  can  be  written  at  the  present 
time  is  that  given  in  Table  IV. 

That  table  shows  values  for  railway  bridges  and  trestles 
adopted  by  the  American  Railway  Engineering  Associa- 
tion. 

As  in  the  case  of  tension, the  compressive  resistances  across 
the  grain  are  but  small  fractions  of  those  with  the  grain. 
Values  are  given  for  columns  under  15  diameters  in  length 
for  the  reason  that  such  columns  fail  essentially  by  com- 


434 


COMPRESSION. 


[Ch.  VIII. 


pression  and  without  the  bending  which  characterizes  long 
columns.     The  table  is  one  of  great  practical  value. 

TABLE  IV. 
TIMBER  IN  COMPRESSION. 


Kind  of  Timber. 

Unit  Stresses  in  Lbs.  per  Sq.  In. 

Perpendicular  to 
the  Grain. 

Parallel  to 
the  Grain. 

Working  Stresses 
for  Columns. 

Elastic 
Limit. 

Working 
Stress. 

Mean 
Ult. 

Working 
Stress. 

Length 
Under 
15  Xd. 

Length  Over  15  Xd. 

Douglas  fir  .... 
Longleaf  pine  .  . 
Shortleaf  pine.  . 
White  pine  .... 
Spruce  

630 
520 
340 
290 
370 

310 
260 
170 
150 

180 
150 

220 

22O 
150 
170 
230 
450 

3,600 
3,800 
3400 

3,000 
3,200 
2,600* 
3,200* 

3,500 
3,300 
3,900 
2,800 
3,500 

1,200 
1,300 
1,100 
I,OOO 
1,100 

800 

1,000 

1,200 
9OO 
1,100 
900 
1,300 

900 

975 
825 
750 
825 
600 
750 

900 
675 
825 
675 
975 

,200(1  —l/6od) 
,300(1  -l/6od) 
,ioo(i-//6o</) 
,000(1  —l/6od) 
,100(1  —l/6od) 
8oo(i-//6od) 
1,000(1  —l/6od) 

I.200(l-//6od) 

900(1  -l/6od) 
i,ioo(i-//6od) 
900(1  -l/6od) 
1,300(1  —l/6od) 

Norway  pine.  .  . 
Tamarack 

Western   Hem- 
lock       

440 
400 
340 
470 
920 

Redwood.    .  .  . 

Bald  cypress.  .  . 
Red  cedar  
White  oak  

Unit  stresses  are  for  green  timber  and  are  to  be  used  without  increasing 


the  live  load  stresses 
timbers. 

In  the  formulas  given  for  columns,    / 
d  =  least  side  or  diameter,  in  inches. 


for    impact.      Values  noted   *  are  partially  air-dry 
length  of  column,  in  inches,  and 


CHAPTER   IX. 
RIVETED  JOINTS  AND  PIN  CONNECTION. 

Art.  71.— Riveted  Joints. 

ALTHOUGH  riveted  joints  possess  certain  characteristics 
under  all  circumstances,  yet  those  adapted  to  boiler  and 
similar  work  differ  to  some  extent  from  those  found  in  the 
best  riveted  trusses.  The  former  must  be  steam-  and  water- 
tight, while  such  considerations  do  not  influence  the  design 
of  the  latter,  consequently  far  greater  pitch  may  be  found 
in  riveted-truss  work  than  in  boilers.  Again,  the  peculiar 
requirements  of  bridge  and  roof  work  frequently  demand 
a  greater  overlap  at  joints  and  different  distribution  of 
rivets  than  would  be  permissible  in  boilers. 

Kinds  of  Joints. 

Some  of  the  principal  kinds  of  joints  are  shown  in  Figs,  i 
to  6.  Fig.  i  is  a  " lap-joint"  single-riveted;  Fig.  2  is  a 
"lap-joint"  double-riveted;  Fig.  3  is  a  "butt-joint"  with 
a  single  butt-strap  and  single-riveted;  whi1e  Figs.  4,  5,  and 
9  are  "butt-joints"  with  double  butt-straps,  Fig.  4  being 
single-riveted,  while  the  others  are  double-riveted.  Fig.  5 
shows  zigzag  riveting,  and  Fig.  6  chain  riveting.  All  these 
joints  are  designed  to  resist  tension  and  to  convey  stress 
from  one  single  thickness  of  plate  to  another.  Two  or 

435 


436 


RIVETED  JOINTS  AND   PIN  CONNECTION.          [Ch.  IX. 

n 


OOO 


000 

ooo 


-T 


ft 

>-*• 


o 


000 


FlG.    I. 


FIG.  2. 


FIG.  3. 


ooo 

ooo 


3 


ooo 


ooo 

ooo 


> 


FIG.  6. 


1 

E 

IP 

p 

J 

Q---ME 

)        O 

c 
/ 

P 

) 
i 

'  / 

^ 

i 

Qp-Q  C 

)  O  O 

( 

,                 -n 

-T' 

CI 

O   -  i  G 

•\A           /^ 

( 

)      B 

P 

'              W 

B              * 

C 

p 

D 

•* 

t' 

1 

> 

FIG.  7. 


FIG.  8. 


FIG.  9. 


Art.  72.]        DISTRIBUTION   OF  STRESS  IN  RIVETED  JOINTS.       437 

three  other  joints   peculiar  to  bridge  and   roof  work  will 
hereafter  be  shown. 

In  the  cases  of  bridges  and  roofs  these  " butt-straps" 
are  usually  called  " cover-plates." 

Art.  72.— Distribution  of  Stress  in  Riveted  Joints. 

Bending  of  the  Plates. 

In  order  that  rivets,  butt-straps  or  cover-plates  and 
different  parts  of  the  main  plates  may  take  their  proper 
stresses,  an  accurate  adjustment  of  these  different  parts  to 
the  external  forces  or  loads  must  be  attained ;  but  all  shop 
work  is  necessarily  more  or  less  imperfect  and  the  varying 
stresses  at  different  parts  of  the  joint  produce  at ,  least 
elastic  deformations  so  that  the  requisite  conditions  for  a 
proper  distribution  of  stresses  as  computed  cannot  be  main- 
tained. The  precise  amount  of  stress,  therefore,  carried 
by  each  rivet,  cover-plate  or  other  part  of  the  joint  in- 
cluding the  main  plates  cannot  be  computed.  By  means 
of  reasonable  assumptions,  however,  and  by  the  introduc- 
tion of  factors  or  coefficients  determined  by  the  actual 
testing  of  riveted  joints,  simple  and  sufficiently  accurate 
formulas  for  all  engineering  purposes  may  be  established. 

The  shafts  of  the  rivets  of  any  joint  compress  or 
bear  against  the  walls  of  the  rivet  holes  in  the  transference 
of  loading  from  one  main  plate  to  the  other.  This  con- 
dition will  necessarily  subject  the  metal  on  either  side  of 
the  hole  and  adjacent  to  it  to  a  higher  degree  of  tension 
than  the  metal  midway  between  two  neighboring  holes. 
This  makes  the  average  intensity  of  stress  over  the  minimum 
section  of  either  the  main  plate  or  the  cover-plate  materi- 
ally less  than  the  maximum  intensity  at  or  near  the  wall 
of  the  rivet  hole.  On  the  other  hand,  the  removal  of  the 
metal  for  the  rivet  holes  makes  that  part  of  the  plates 


438  RIVETED  JOINTS  AND  PIN  CONNECTION.         [Ch.  IX. 

between  two  consecutive  holes  at  right-angles  to  the  direc- 
tion of  loading  a  "  short  "  specimen  with  a  higher  ultimate 
resistance  than  a  long  specimen. 

Again  let  Fig.  8,  like  Fig.  2  of  the  preceding  article, 
represent  a  longitudinal  section  of  a  double  riveted  lap- 
joint,  the  thicknesses  of  the  two  plates  being  t  and  tr.  The 
two  opposite  loads  P  would  produce  a  bending  moment 
about  an  axis  at  right-angles  to  the  plane  of  section  of 

t+t' 

P .     Usually  the  two  thicknesses  of  plate  are  the  same 

2 

making  t  the  lever  arm  of  the  couple.  This  moment  causes 
bending  in  the  plates  in  the  vicinity  of  A  and  B  of  equal 
amount  and  the  bending  intensities  of  stresses  may  be  com- 
puted in  the  usual  manner  if  the  joints  were  not  distorted 
so  as  to  change  the  lever  arm  of  the  couple.  As  the  load 
is  increased,  however,  the  joint  tends  to  take  the  shape 
shown  in  Fig.  9,  the  two  plates  tending  to  pull  into  the 
same  straight  line,  making  it  impossible  to  compute  accu- 
rately the  bending  moment.  It  is  sufficient,  however,  to 
recognize  this  condition  of  flexure  in  the  joint. 

This  eccentric  action  of  the  load  P  produces  also  the 
same  bending  moment  in  the  rivets  of  the  joint,  in  the 
aggregate,  as  that  impressed  upon  the  plates.  The  assumed 
bending  moment  carried  by  each  rivet  will  be  the  moment 

t+t' 

p or  Pt  divided  by  the  number  of  rivets  in  the  joint. 

2 

This  bending  moment  is  seldom  or  never  computed  for 
rivets  but  it  is  always  computed  in  the  design  of  pins  of 
a  pin-connected  truss  bridge. 

For  all  these  reasons  and  others  shortly  to  be  considered 
it  is  obvious  that  if  a  riveted  joint  of  any  type  be  tested  to 
destruction,  it  is  essentially  impossible  to  compute  accu- 
rately what  the  intensity  of  stress  will  be  in  any  part  of 
it  at  any  stage  of  loading.  Such  tests,  however,  yield  most 


Art.  72.]     DISTRIBUTION  OF  STRESS  IN  RIVETED  JOINTS.          439 

valuable  empirical  quantities  to  be  used  in  formulae  to  be 
established  and  without  which  it  would  be  essentially 
impossible  to  design  a  riveted  joint  in  a  rational  manner. 

Although  these  considerations  are  based  upon  the  charac- 
teristics of  a  double-riveted  lap-joint,  they  apply  to  all 
riveted  joints  of  any  type  whatever.  If  the  butt-joint  with 
double  cover-plates  shown  in  Fig.  5  of  the  preceding 
article  be  considered,  it  will  be  clear  at  once  that  if  a  line 
be  drawn  centrally  through  the  section  of  the  two  main 
plates,  each  half  of  the  actual  joint  will  be  divided  into 
two  equal  double-riveted  lap-joints  in  each  of  which  the 
plates  will  be  subjected  at  least  approximately  to  the  same 
condition  of  stress  as  that  found  in  connection  with  Fig.  9 
and  the  bending  of  the  rivets  will  be  precisely  the  same. 
There  will  be,  however,  no  bending  of  the  main  plates. 

The  special  form  of  joint  shown  in  Fig.  7,  which  has 
come  to  be  much  used,  will  also  have  its  parts  subjected 
to  the  same  general  condition  of  stresses  including  the  bend- 
ing of  rivets  and  main  plates. 

It  is  clear  that  the  bending  of  the  plates  illustrated  in 
Fig.  9  will  increase  with  their  thickness. 


Net  Section  of  Plates 

The  net  section  of  any  main  plate  or  cover-plate  in  a 
riveted  joint  is  the  gross  section  along  any  transverse  line 
of  rivets  less  the  metal  taken  out  by  the  rivet  holes.  In 
Fig.  2  of  the  preceding  article,  the  net  section  of  either  main 
plate  will  be  its  gross  section  less  three  rivet  holes.  The 
pitch  p  of  the  rivets  in  any  transverse  line  of  rivet  holes 
in  a  riveted  joint  is  the  distance  between  the  centres  of  two 
consecutive  rivets  as  shown  in  Fig.  7.  In  the  centre  line 
of  rivets  in  that  figure,  the  pitch  is  one-half  that  in  the  outer 
line.  The  net  section  of  any  plate,  therefore,  per  rivet  will 


440  RIVETED   JOINTS  AND   PIN  CONNECTION.          [Ch.  IX. 

be  (p—d)t,  d  being  the  diameter  of  the  rivet  hole  and  t 
the  thickness  of  the  plate.  If  n  is  the  number  of  rivets 
in  one  main  plate  and  if  q  is  the  number  of  rows  of  rivets 

in  it,  then  the  number  of  rivets  in  each  row  will  be  -  and  the 

8 
total  net  section  along  any  transverse  row  of  rivets  will 


Bending  of  the  Rivets. 

It  has  already  been  seen  that  the  rivets  of  any  riveted 
joint  are  subjected  to  bending.     It  is  assumed  that  the 

t+tf 

total   bending   moment,    M=P-    -orM=Pt   is    divided 

2 

uniformly  among  all  the  rivets  of  the  joint.      Hence  the 
bending  moment  to  which  a  single  rivet  is  subjected  is 

M     kAd  . 


in  which  A  is  the  area  of  cross  section  of  one  rivet 
and  k  the  greatest  intensity  of  tension  or  compression  in 
the  extreme  fibre  due  to  bending.  By  introducing  in  eq. 
(i)  the  values  of  M  already  used,  eqs.  (2)  and  (3)  at  once 
result. 


if /=*', 


nAd 


This  equation  is  approximate  because  it  is  virtually 
assumed  that  the  pressure  on  the  rivet  is  uniformly  dis- 


Art.  72.]       DISTRIBUTION  OF  STRESS  IN   RIVETED   JOINTS.          441 

tributed  along  its  axis.*  This  is  a  considerable  deviation 
from  the  truth,  particularly  as  failure  is  approached.  The 
true  bending  moment  is  much  less  than  that  given  by 
eq.  (i)  after  the  rivet  has  deflected  a  little. 

When  the  joint  takes  the  position  shown  in  Fig.  9,  it 
is  clear  that  the  rivet  is  also  subject  to  some  direct  tension. 

The  Bearing  Capacity  of  Rivets. 

There  is  a  very  high  intensity  of  pressure  between  the 
shaft  of  the  rivet  and  the  wall  of  the  hole.  This  intensity 
is  not  uniform  over  the  surface  of  contact,  but  has  its 
greatest  value  at,  or  in  the  vicinity  of,  the  extremities  of 
that  diameter  lying  in  the  direction  of  the  stress  exerted 
in  the  plate.  At  and  near  failure  this  intensity  may  be 
equal  to  the  crushing  resistance  of  the  material  over  a  con- 
siderable portion  of  the  surface  of  contact. 

The  intricate  character  of  the  conditions  involved  ren- 
ders it  quite  impossible  to  determine  the  law  of  the  dis- 
tribution of  this  pressure.  The  bending  of  the  rivets  under 
stress  tends  to  a  concentration  of  the  pressure  near  the 
surface  of  contact  of  the  joined  plates,  while  the  unavoid- 
ably varying  "fit"  of  the  rivet  in  its  hole,  even  in  the  best 
of  work,  throws  the  pressure  towards  the  front  portion  of 
the  surface  of  the  rivet  shaft.  The  intensity  thus  varies 
both  along  the  axis  and  around  the  circumference  of  the 
rivet. 

If  any  arbitrary  law  is  assumed,  the  greatest  intensity 
of  pressure  is  easily  determined.  Such  laws,  however,  are 
mere  hypotheses  and  possess  no  real  value.  All  that  can 
be  done  is  to  determine,  by  experiment,  the  mean  safe 

*  In  accordance  with  this  assumption,  strictly  speaking,  \t  (thickness  of 
main  plate)  should  be  taken  instead  of  i  in  the  sum  (/-h/')  in  the  above 
formulae  for  bending,  when  applied  to  the  double  butt-joint,  Figs.  5  and  6. 


442  RIVETED  JOINTS  AND  PIN  CONNECTION.         [Ch.  IX. 

working  intensity  on  the  diametral  plane  of  the  rivet  which 
is  equivalent  to  a  fluid  pressure  of  the  same  intensity  against 
its  shaft. 

Thus,  if  /  is  this  mean  (empirically  determined)  intensity, 
d  the  diameter  of  the  rivet,  and  t  the  thickness  of  the  plate, 
the  total  pressure  carried  by  one  rivet  pressing  against  one 
plate  is 

' 


Bending  of  Plate  Metal  in  Front  of  Rivets. 

In  addition  to  the  bending  of  the  plates  of  a  riveted  joint 
about  an  axis  parallel  to  the  plates  and  at  right  angles  to 
the  direction  of  loading,  there  is  further  bending  of  the 
metal  immediately  in  front  of  a  rivet  about  an  axis  parallel 
to  the  axis  of  the  rivet.  If  a  rivet,  such  as  A,  Fig.  7,  be 
considered,  the  metal  on  that  side  of  the  hole  nearest  to 
the  line  BC  will  be  in  the  condition  approximately  of  a  beam 
fixed  at  each  end  of  the  diameter  of  the  hole  parallel  to  BC, 
the  bearing  load  jdt  being  the  load  resting  upon  it  and 
assumed  to  be  uniformly  distributed  over  the  span  d. 
Manifestly  the  depth  of  this  beam  is  not  uniform,  but  it 

is  assumed  to  have  a  depth  h  —  ,  Fig.  7,  throughout  the 

2 

span  d.  If  t  is  the  thickness  of  the  plate,  p  the  pitch  of  the 
rivets  and  T  the  mean  intensity  of  tension  between  the 
rivet  holes,  the  load  on,  this  beam  will  be  (p—d)Tt  and  the 
moment  of  inertia  of  the  cross-section  will  be 


12 


It  will  be  shown  in  the  chapter  on  bending  that  k  may 

here  be  taken  at  -T. 
2 


Art.  72.]     DISTRIBUTION  OF  STRESS  IN  RIVETED  JOINTS.          443 

In  Art.  30  the  moments  at  the  centre  and  end  of  a  span 
fixed  at  each  end  and  uniformly  loaded  were  shown  to  be 
T^  of  the  load  into  the  span  for  the  end  moments  and  ^ 
of  the  load  into  the  span  for  the  centre  moment. 

Hence,  by  the  usual  formulae, 


12 


HM    '  ' 

Sd  .  (5) 


Shearing  of  Rivets. 

The  shearing  of  the  rivets  in  a  riveted  joint  takes  place 
in  the  plane  of  the  surface  of  contact  between  any  two 
plates  tending  to  move  in  opposite  directions.  In  Fig.  8 
the  plane  of  shear  would  be  the  surface  of  contact  between 
the  main  plates  A  and  B,  and  in  Fig.  7  on  both  sides  of  the 
main  plate,  F,  i.e.,  between  the  main  plates  E  and  F  and 
at  the  surface  of  contact  between  the  main  plate  F  and  the 
bent  cover-plate  D.  It  is  assumed  that  the  total  shear  is 
divided  uniformly  between  all  the  shear  sections  of  the 
rivets  so  that  if  n  were  the  total  number  of  rivets  carrying 
the  load  P  and  if  d  be  the  diameter  of  the  rivet  while  5  is 
the  intensity  of  shearing  stress  in  the  normal  sections  of 
the  rivets,  there  would  result  for  single  shear  the  expression 
P  =n. 7854^5.  The  rivets  shown  in  Fig.  8  and  Figs,  i,  2, 
and  3  of  the  preceding  article  are  in  single  shear.  If  each 
rivet  must  be  sheared  at  two  normal  sections  in  order 
that  the  joint  may  fail  (by  shear),  as  in  Figs.  4,  5,  and  6 
of  the  preceding  article,  the  rivets  are  said  to  be  in  double 
shear.  In  the  latter  case  in  the  preceding  expression  2n 
must  be  written  for  n  for  all  rivets  in  double  shear.  In 


444  RIVETED  JOINTS  AND   PIN  CONNECTION.         [Ch.  IX. 

Fig.  7  the  two  lower  rows  of  rivets  are  in  double  shear  and 
the  upper  row  in  single  shear. 

In  Fig.  8  and  in  Figs.  5  and  6  of  the  preceding  article, 
each  row  of  rivets  is  assumed  to  take  half  the  total  load 
carried  by  the  joint.  That  condition,  if  the  cover-plates  of 
Figs.  5  and  6  are  of  half  the  thickness  of  the  main  plates, 
makes  the  intensity  of  stress  the  same  in  the  main  plate 
and  in  the  two  covers  between  the  two  rows  of  rivets  on 
either  side  of  the  joint.  If,  however,  the  thickness  of  the 
cover-plate  is  greater  than  one-half  the  thickness  of  the  main 
plate,  as  is  always  the  case  in  such  joints,  then  if  each  row 
of  rivets  carries  half  the  load,  the  intensity  of  stress  in  the  two 
covers  between  each  two  rows  of  rivets  will  be  less  than  in 
the  main  plate  causing  the  rate  of  stretch  in  the  latter  to  be 
greater  than  in  the  former.  This  condition,  would  throw 
more  than  half  the  load,  as  shear,  on  the  outer  row  of  rivets. 
In  other  words,  the  tendency  will  be  to  make  the  stretch 
of  the  plates  within  the  joint  added  to  the  distortion  due 
to  bending  and  shearing  of  the  rivets  equal  to  each  other 
between  each  pair  of  rows  of  rivets  parallel  to  the  joint 
line  between  the  main  plates.  If  again  there  are  three  or 
more  rows  of  rivets  on  either  side  of  an  abutting  joint, 
there  will  be  a  corresponding  tendency  to  overload  the 
outer  rows  of  rivets  and  relieve  those  nearest  the  centre 
or  abutting  line  of  the  joint.  There  are  further  conditions 
in  addition  to  those  already  discussed,  militating  against 
perfect  uniformity  in  the  stress  conditions  of  the  complete 
joint.  It  is  impossible,  however,  to  make  allowance  for 
these  complicated  and  more  or  less  obscure  stress  con- 
ditions in  the  operations  of  design  or  development  of 
formulae.  Hence,  as  already  indicated,  the  usual  assump- 
tions of  uniformity  in  the  three  principal  methods  of  failure 
of  riveted  joints  are  made  leaving  the  working  stresses  to 
be  determined  by  the  results  of  tests  of  actual  joints. 


Art.  73.]  DIAMETER  AND  PITCH   OF  RIVETS.  445 

Art.  73.—  Diameter  and  Pitch  of  Rivets  and  Overlap  of  Plate. 
Distance  between  Rows  of  Riveting. 

Diameter  of  Rivets. 

The  diameter  of  rivet  may  at  least  approximately  be 
expressed  in  terms  of  the  thickness  of  the  plate  which  it 
pierces.  There  are  various  arbitrary  or  conventional 
rules  based  upon  this  method  of  determining  the  rivet 
diameter.  If  the  unit  is  the  inch,  the  diameter  d  may  be 
expressed  as  ranging  between  the  two  following  values 
for  ordinary  thicknesses  of  plate: 

d  =  -7$t   + 


.375,) 
-375,  I 


in  which  t  is  the  thickness  of  the  plate.  Unwin  gives  the 
following  expression  for  the  diameter  of  somewhat  different 
i'orm  from  that  which  precedes: 

d  =  i.2VT.       ......     (2) 

Neither  of  the  preceding  expressions  can  be  applied 
for  all  thicknesses  of  plates.  If  the  thickness  is  great, 
those  expressions  make  the  diameter  of  the  rivet  too  large, 
the  diameter  rarely  exceeding  i  inch  even  for  the  heaviest 
plates.  The  application  of  eq.  (i)  to  different  thicknesses 
of  plates  will  give  the  following  diameters  of  rivets  ex- 
pressed by  the  nearest  TV  in.  : 

/  d 

i  in.  T9B  in. 

I  i 

i  I 

f  if 

i 

*  it 

i  iA 


446  RIVETED  JOINTS  AND  PIN  CONNECTION.         [Ch.  IX. 

In  structural  work  for  ordinary  thicknesses  of  metal 
the  prevailing  diameters  of  rivets  are  f  in.  and  f  in.  For 
light  work,  such  as  sidewalk  railings  or  light  highway 
construction,  rivets  as  small  as  J  in.  or  f  in.  in  diameter 
are  used.  On  the  other  hand,  i  to  if -inch  rivets  are 
employed  for  specially  heavy  sections. 

Pitch  of  Rivets. 

It  is  possible  to  determine  the  pitch  of  rivets  approxi- 
mately by  an  equation  expressing  equality  between  the 
tensile  resistance  of  the  net  section  between  two  adjacent 
rivets  and  the  shearing  or  bearing  capacity  of  a  single  rivet, 
but  it  is  scarcely  practicable  to  proceed  in  that  manner 
as  a  rule.  Again,  the  pitch  will  vary  to  some  extent  with 
the  number  of  lines  of  riveting  on  either  side  of  the  joint. 
In  single-riveting  the  pitch  must  be  less  than  in  double- 
or  other  multiple-riveting.  In  boiler  or  other  similar 
riveting,  also,  the  pitch  must  be  usually  less  than  in  struc- 
tural work,  as  questions  of  steam-  and  water-tightness  or 
other  similar  considerations  are  involved  in  the  former 
class  of  joints.  Finally,  the  pitch  will  also  obviously 
depend  largely  upon  the  thickness  of  plates.  In  single- 
riveting  for  comparatively  thin  plates  the  following  rela- 
tion may  be  taken,  p  being  the  pitch  in  inches : 

/?  =  1.75  in.  to  2.25  in (3) 

For  comparatively  thick  plates  in  single-riveting  the  follow- 
ing relation  may  holil: 

£  =  2.375  in.  to  3  in (4) 

In  double-riveting,  p  and  /  still  being  the  pitch  and  thick- 
ness respectively,  the  following  relation  may  be  taken  for 
comparatively  thin  plates: 


Art.  73.]  DIAMETER  AND  PITCH  OF  RIVETS.  447 

£=2.6875  in.  to  3.25  in (5) 

Again,  for  comparatively  thick  plates  in  double-riveting, 

£=3-375  in.  to  3.75  in (6) 

The  values  given  by  eqs.  (3)  to  (6)  are  for  boiler  or 
other  similar  work. 

In  structural  work  the  pitch  of  rivets  is  seldom  less  than 
about  three  times  the  diameter  of  the  rivet,  and  it  is  fre- 
quently specified  not  to  exceed  sixteen  times  the  thickness 
of  the  thinnest  plate  pierced  by  the  rivet. 

Overlap  of  Plate. 

The  overlap  of  a  plate,  h  in  Fig..  2,  Art.  71,  in  a  riveted 
joint  is  the  distance  from  the  edge  of  the  plate  to  the  centre 
line  of  the  nearest  row  of  rivets.  This  distance,  like  other 
elements  of  riveted  joints,  will  depend  somewhat  upon 
the  thickness  of  the  plate  as  well  as  the  diameter  of  rivet 
and  other  similar  considerations.  It  is  a  common  practice 
to  make  the  overlap  not  less  than  about  1.5^,  d  being  the 
diameter  of  the  rivet.  Occasionally  in  riveted  joints  it 
is  made  a  little  less,  but  i .  5  times  the  diameter  of  the  rivet 
is  about  as  small  as  the  overlap  should  be  made.  Some- 
times J  in.  is  added  to  the  preceding  value  of  the  overlap. 

The  width  of  overlap  (h)  may  also  be  determined  in 
terms  of  d  by  the  aid  of  eq.  (u)  of  Art.  72.  Since  the  load 
on  the  rivet  is  represented  by  (p  —  d)Tt,  p  must  be  taken 
in  terms  of.  d  for  a  single-riveted  joint,  in  which  p  =  2^d  to 
i\d.  As  a  margin  of  safety,  and  as  it  will  at  the  same 
time  simplify  the  resulting  expression,  let  p  =$d. 

Eq.  (5)  of  Art.  72  then  gives,  in  confirmation  of  the 
preceding  rule,  /*  =  i.3id* (7) 

*In  consequence  of  the  direct  tension  in  the  metal  on  either  side  of  the 
rivet  this  value  of  h  should  be  increased,  i.e.,  to  perhaps  1.5^. 


448  RIVETED  JOINTS  AND  PIN  CONNECTION.          [Ch.  IX. 

Experience  has  shown  that  this  rule  gives  ample  strength, 
and  is  about  right  for  calking  in  boiler  joints. 

It  is  to  be  remembered  that  the  preceding  conventional 
rules  for  the  diameter  of  rivet,  pitch,  and  overlap  of  plate 
are  necessarily  to  a  large  extent  conventional  or  approxi- 
mate, and  in  special  cases  they  cannot  be  applied  with 
mathematical  exactness.  As  practical  rules,  however, 
they  are  sufficiently  close  to  give  good  general  ideas  of 
those  features  of  riveted  joints. 

Distance  between  Rows  of  Riveting. 

The  distance  between  the  rows  of  riveting  is  not  .susceptible 
of  accurate  expression  by  formulas,  although  the  considera- 
tions involved  in  the  establishment  of  eq.  (n)  of  Art.  72 
would  lead  to  an  approximate  value.  It  is  evident,  how- 
ever, that  this  distance  should  never  be  as  small  as  h. 
Apparently,  in  more  than  double-riveted  joints,  this  dis- 
tance should  increase  as  the  centre  line  of  the  joint  is 
receded  from,  in  consequence  of  the  bending  action  of  the 
rivet.  There  are  other  reasons,  however,  besides  that  of 
inconvenience,  why  such  a  practice  is  not  advisable. 

In  chain  riveting  the  distance  between  the  centre  lines  of 
the  rows  of  rivets  may  be  taken  equal  to  the  pitch  in  a  single- 
riveted  joint,  or,  as  a  mean,  at  2.5  the  diameter  of  a  rivet. 

In  zigzag  riveting  (Fig.  5)  this  distance  may  be  taken  at 
three  quarters  its  value  for  chain  riveting. 

Art.  74. — Lap-joints,  and  Butt-joints  with  Single  Butt-strap  for 

Steel  Plates. 

A  butt- joint  with  single  butt-strap,  similar  to  that  shown 
in  Fig.  3,  Art.  71,  is  really  composed  of  two  lap-joints  in 
contact,  since  each  half  of  the  butt-strap  or  cover-plate 


Art.  74.]  LAP-JOINTS  AND  BUTT-JOINTS.  449 

with  its  underlying  main  plate  forms  a  lap-joint.      It  is 
unnecessary,  therefore,  to  give  it  separate  treatment. 

From  these  considerations  it  is  clear  that  the  thickness 
of  the  butt-strap  or  cover-plate  should  be  at  least  equal  to 
that  of  the  main  plate ;  it  is  usually  a  little  greater. 
Let  t=  thickness  of  plates; 
d  =  diameter  of  rivets ; 
p=  pitch  of  rivets    (i.e.,    distance  between  centres 

in  the  same  row) ; 
T  =  mean  intensity  of  tension  in  net  section  of  plates 

between  rivets; 

T'=mean  intensity  of  tension  in  main  plates; 
/  =  rnean  intensity  of  pressure  on  diametral  plane 

of*  rivet ; 

5=  mean  intensity  of  shear  in  rivets; 
n  =  number  of  rivets  in  one  main  plate ; 
q  =  number  of  rows  in  one  main  plate ; 
h=\ap  as  shown  in  Fig.  2,  Art.  71. 

If  all  the  dimensions  are  in  inches,  then  T,  Tf,  f,  and  5 
are  in  pounds  per  square  inch. 

The  starting-point  in  the  design  of  a  joint  is  the  thickness 
t  of  the  plate.  The  rivet  diameter  may  then  be  expressed 
in  terms  of  t,  and  the  pitch  in  terms  of  the  diameter.  Such 
rules,  like  those  given  in  Art.  72,  may  be  useful  within 
a  certain  range  of  application,  but  they  cannot  be  depended 
upon  in  all  cases. 

The  thickness  t  of  boiler-plate  depends  upon  the  internal 
pressure,  and  is  to  be  determined  in  accordance  with  the 
principles  laid  down  in  Art.  39,  after  having  made  allowance 
for  the  metal  punched  out  at  the  holes  to  find  the  net 
section. 

In  truss  work  the  thickness  depends  upon  the  amount 
of  stress  to  be  carried,  and  the  same  allowance  is  to  be 
made  for  rivet-holes  in  finding  the  net  section. 


450  RIVETED  JOINTS  AND  PIN  CONNECTION.          [Ch.  IX. 

The  relation  existing  between  T  and  Tf  is  shown  by  the 
following  equations: 


or 

£_fci*-i  d 

T        P  f 


In  order  that  the  joint  may  be  equally  strong  in  refer 
ence  to  all  methods  of  failure,  the  following  series  of  equali 
ties  must  hold: 

-tpT'=-t(p 

.'.  tpT'  =t(p-d)T=qfdt  =  0.7854^5.       .     .      (2) 

It  is  probably  impossible  to  cause  these  equalities  to 
exist  in  any  actual  joint,  but  none  of  the  intensities  T',  T, 
/,  or  5  should  exceed  a  safe  working  value. 

The  method  of  failure  by  tearing  through  the  gross 
section  of  the  main  plate  is  practically  impossible  under 
ordinary  circumstances,  and  it  is  neglected  in  designing 
riveted  joints.  This  neglect  is  expressed  by  dropping 
the  first  member  of  eq.  (2)  and  thus  reaching  eq.  (3)  : 

.....      (3) 


This  equation  shows  that  the  usual  design  of  a  riveted 
joint  must  provide  against  failure  in  three  principal  ways  : 

1.  Tearing  through  the  net  section  of  the  plate. 

2.  Compression  of  the  metal  where  the  rivets  bear  against 

the  plate. 

3.  Shearing  of  the  rivets. 

Although   these    are    the   three    principal    methods    of 


Art.  74.]  LAP-JOINTS   AND   BUTT-JOINTS.  451 

failure  of  riveted  joints,  whatever  may  be  their  type  or 
form,  the  proper  design  of  such  joints  should  be  so  per- 
formed as  to  afford  provision  also  against  the  secondary 
stresses  caused  by  rivet  bending,  bending  of  the  plates,  and 
other  indirect  influences  discusssd  in  preceding  articles. 
This  latter  end  is  attained  by  determining  the  empirical 
intensities  T,  /,  and  5  of  eq.  (3)  by  testing  to  failure  actual 
riveted  joints  in  which  those  secondary  stresses  exist.  In 
that  manner  the  design  against  the  three  principal  methods 
of  failure,  described  above,  will  also  afford  provision  against 
the  secondary  or  indirect  stresses  of  rivet  and  plate  bend- 
ing or  other  similar  conditions.  The  determination  of  the 
intensities  T,  /,  and  5  by  tests  of  actual  riveted  joints  will 
be  fully  shown  in  the  following  articles. 

It  may  be  stated  here,  however,  that  an  approximate 
relation  between  the  ultimate  intensities  of  resistance  to 
shear  and  tension  for  steel  has  been  used  in  engineering 
practice  in  accordance  with  which 

S  =  .7$T  ......    ,     .     (4) 

It  will  be  found  hereafter  that  /  may  be  taken  at  least 
1.25  T.  If  these  values  be  substituted  in  the  third  and 
fourth  members  of  eq.  (3)  in  which  q  =  2,  there  will  result 

......     (5) 


This  value  of  d  is  too  large  for  thick  plates. 

The  rivet  diameter,  therefore,  for  steel  plates  may  be  said 
to  vary  from  2t  for  thin  plates  to  i.6t  for  thick  ones,  with 
a  maximum  diameter  of  ij  to  i^  inches.  The  distance 
between  the  centre  lines  of  the  rows  of  rivets  may  be  taken 
at  2.5^  to  ^d  for  chain  riveting  and  three  fourths  of  that 
distance  for  zigzag  riveting. 

The  best  designed  single  -riveted  lap-joints  give  from 
55  to  64  per  cent,  the  strength  of  the  solid  plates. 


452        .  RIVETED  JOINTS  AND  PIN  CONNECTION.  [Ch.  IX. 

Well-designed  double  -  riveted  lap  -  joints  should  give 
Irom  65  to  75  per  cent,  the  resistance  of  the  solid 
plate. 

Equally  well-constructed  treble-  and  quadruple-riveted 
joints  should  have  an  efficiency  of  70  to  80  per  cent,  of  the 
solid  plate. 

It  is  therefore  seen  that  there  is  little  economy  in  more 
than  double-riveting  ordinary  joints. 

Art.  75. — Steel  Butt-joints  with  Double  Cover-plates. 

Butt-joints  with  double  butt-straps  or  covers  differ  in 
two  respects,  and  advantageously,  from  lap-joints  and  butt- 
joints  with  a  single  cover;  i.e.,  in  the  former  the  rivets  are 
in  double  shear  and  the  main  plates  are  subjected  to  no 
bending.  The  cover-plates,  however,  are  subjected  to 
greater  flexure  than  the  plates  of  a  lap-joint,  for  there  is 
no  opportunity  to  decrease  the  leverage  by  stretching.  As 
the  covers  form  only  a  small  portion  of  the  total  material, 
these,  with  economy,  may  be  made  sufficiently  thick  to 
resist  this  tendency  to  failure. 

Let  tr  =  thickness  of  each  cover-plate ;  and  let  the  re- 
maining notation  be  the  same  as  in  Art.  74.  The  intensity 
of  compression  between  the  walls  of  the  holes  in  the  cover- 
plates  and  the  rivets,  and  the  tension  in  the  former,  will 
be  ignored  on  account  of  the  excess  in  thickness  of  the  two 
cover-plates  combined  over  that  of  the  main  plate.  This 
excess  in  thickness  is  required  on  account  of  the  bending 
in  the  covers  noticed  above. 

The  thickness  of  each  cover  should  be  from  f  to  f  the  thick- 
ness of  the  main  plates,  or  t'  =.625  to  .875^. 
»     The  combined  thickness  of  the  covers  will  thus  be  from 
1.25  to  1.75  that  of  the  main  plates. 


Art.  75-1         BUTT-JOINTS   WITH  DOUBLE  COVER-PLATES.  453 

The  four  principal  methods  of  rupture  in  the  main  plate 
will  then  lead  to  the  following  equations,  corresponding  to 
eq.  (2),  Art.  74: 


-t(p-d)T=nfdt 


.       .     .     (i) 

As  in  Art.  74,  and  for  the  reasons  there  given,  the  first 
member  of  eq.  (i)  may  be  omitted,  thus  giving 

t(p-d)T=qfdt  =  i.$>jo&qd2S  .....      (2) 

Tests  of  steel  butt-  joints  with  double  cover-plates  as 
well  as  other  tests  in  bearing  and  tension  in  net  section  of 
plates  make  it  reasonable  to  take  /=  1.257,  with  T  having 
values  from  55,000  to  60,000  pounds  per  square  inch  for 
thick  plates  to  perhaps  65,000  to  70,000  pounds  per  square 
inch  for  thin  plates. 

With  this  value  of  /,  and  q  =  2,  the  first  and  second 
members  of  eq.  (2)  give  for  double-riveted  butt-joints  with 
two  covers; 

P=3-$d     .     .     .....     (3) 

If  the  same  value  of  /  be  preserved,  there  will  result  for 
single-riveted  butt-joints  with  two  covers 


(4) 


If,  as  in  the  preceding  article,  there  be  taken  5  =  . 
and  /  =  1.257,  the  second  and  third  members  of  eq.    (2) 
give 

d  =  i.o6t  ......     ...      (5) 

This  value  of  the  rivet  diameter  is  too  small  for  thin  plates, 
but  about  right  for  thick  plates. 


454  RIVETED  JOINTS  AND  PIN  CONNECTION.          [Ch.  IX. 

Double-riveted  butt-joints  designed  in  accordance  with 
the  foregoing  deductions  should  give  a  resistance  ranging 
from  65  to  75  per  cent,  of  that  of  the  solid  plate. 

Single-riveted  joints  will  give  an  efficiency  somewhat 
less;  perhaps  from  60  to  65  per  cent. 

It  is  to  be  supposed,  in  applying  the  rules  just  established, 
that  all  steel  plates  are  drilled  or  punched  and  reamed.  • 

As  in  the  preceding  cases,  the  distance  between  the 
centre  lines  of  the  rows  of  rivets  may  be  taken  at  2.5  to  $d 
for  chain  riveting,  and  three  quarters  that  distance  for 
zigzag. 

Art.  76. — Tests  of  Full-size  Riveted  Joints. 

There  have  not  been  many  tests  of  full-size  riveted 
joints  of  either  iron  or  steel,  and  those  which  have  been 
made  seldom  include  such  heavy  steel  plates  as  are  now 
frequently  employed  both  in  boiler  work  and  for  structural 
purposes.  The  most  valuable  tests  avai1able  and  with  the 
greatest  range  in  size  of  r  vet  and  thickness  of  plate  are 
those  which  have  been  made  at  the  U.  S:  Arsenal,  Water- 
town,  Mass.  The  results  shown  in  Table  I  were  taken 
from  "Senate  Ex.  Doc.  No.  i,  47th  Congress,  2d  Session," 
while  those  in  Table  II  are  taken  from  "  Senate  Ex.  Doc. 
No.  5,  48th  Congress,  ist  Session."  The  results  shown  in 
Table  III  are  from  the  same  source  and  are  given  in  the 
"U.  S.  Report  of  Tests  of  Metals  and  Other  Materials" 
for  1896.  The  character  of  plates,  rivets,  and  holes  is 
shown  in  the  tables,  and  the  intensities  of  tension  in  the 
net  sections  of  plates,  compression  or  bearing  on  diametral 
surface,  and  shearing  on  rivets  are  those  which  existed  at 
the  instant  of  failure.  The  bold-face  figures  show  the 
kind  of  failure,  and  when  such  figures  are  found,  for  the 
same  test,  in  two  or  three  columns,  they  show  that  the 
same  two  or  three  kinds  of  failure  took  place  simultaneously. 


Art.  76.]  TESTS   OF  FULL-SIZE  RIVETED   JOINTS. 

TABLE  I. 
RIVETED  JOINTS— IRON  AND  STEEL. 


455 


No. 

Size  of 
Rivet 
and 
Kind. 

Pitch  of 
Rivet. 

Maximum  Stresses, 
Pounds  per  Square  Inch. 

4 

•L 

*o  8 

.§£ 
w 

Remarks. 

Tension 
on  Net 
Area  of 
Plate  (D 

Com- 
pression 
on  Dia- 
metral 
Surface 
(f). 

Shearing 
on 
Rivets 
(5). 

Single-riveted  lap-joints;  J4-inch  iron  plates. 

35 

&£"  iron 

2     ins. 

43,230 

76,140 

34,900 

57-7 

1^4e"  punched  holes. 

36 

sz"      " 

2 

45,520 

82,910 

38,640 

61.4 

41                                 if                                 It 

37 

56"    " 

2 

38,580 

73,260 

34,870 

52.8 

"        drilled       " 

38 

6/8"   11 

2            " 

41,790 

79,36o 

38,660 

57-1 

14                           <l                          14 

39 

i56   " 

52,160 

65,420 

33,420 

60.6 

"      punched     " 

40 

56"    " 

J56  " 

-54,930 

68,890 

35,200 

64.0 

14            '               **                          14 

41 

^"  steel 

2 

49,420 

87,670 

39,640 

65-9 

II                          44                          II 

42 

56" 

2 

47,260 

83,940 

40,610 

63.1 

41                          44                          41 

43 

L&"       " 

Z56  " 

45,890 

78,220 

45,300 

60.3 

94e" 

44 

Ma"     " 

i56  " 

49,720 

84,660 

48,420 

65.5 

II                        II                        M 

45 

7Ae"  iron 

I%6  " 

41,095 

66,778 

44,204 

53-1 

J*j"       drilled 

46 

%«"      ' 

i5Ae  " 

37,500 

60,886 

42,038 

48.3 

Single-riveted  lap-joints;  *4-inch  steel  plates. 

426 

|6"  iron 

2     ins. 

46,340 

82,480 

37,890 

53.2 

1Me"  punched  holes. 

427 

2 

46,010 

81,780 

37,860 

52.8 

" 

436 

%"  steel 

2 

60,250 

107,260 

49,270 

69.  2 

44                          44                          II 

437 

56"     " 

2 

59,240 

105,290 

48,750 

68.0 

44                          14                          44 

428 

%"  iron 

2 

40,950 

77,870 

36,350 

48.2 

drilled       " 

429 

56"     " 

2 

42,370 

80,200 

36,710 

49.6 

44                           4«                         It 

438 

56"  steel 

2 

63,190 

120,160 

56,100 

74-3 

II                          II                         II 

439 

56"    " 

2 

61,310 

1  16,090 

52,460 

71.8 

II                          II                          II 

430 

¥''  11 

t" 

66,860 

90,000 

41,790 

68.8 

II                          It                         44 

" 

70,000 

94,230 

43,750 

72.0 

41                          II                         <« 

47 

%e"   " 

8  " 

62,496 

101,180 

65,220 

69.0 

I/''                          **                         ** 

48 

7Ae"    " 

I%6  " 

58,338 

94,800 

60,382 

64.8 

II                          II                         44 

49 

56"    " 

2 

60,184 

114,603 

52,742 

70.6 

itte" 

50 

56"    " 

2           " 

57,439 

109,650 

50,645 

67.6 

«                      44                     44 

Double-riveted  lap-joints;     J4-inch  plates. 

85 
86 
617 
618 

%«;;  iron 

2     ins. 

2 

38,535 
41,750 
50,592 
49,950 

64,120 
69,710 
42,118 
41,660 

43,110 
41,750 
28,691 
28,660 

60.3!^    drilled     holes. 
|:i|%6"  Punched     « 

Double-riveted  lap-joints;  ^4-inch  steel  plates. 

432 

%"   iron 

2     ins. 

61,510 

54,640 

25,400 

70.4 

1He//  punched  holes. 

433 

yaf 

2 

60,300 

53,7i5 

25,530 

69.4 

( 

434 

56"     " 

2            " 

65,400 

64,600 

30,430 

74-9 

44                        44 

435 

56"     " 

2           " 

64,600 

63,430 

30,430 

74-3 

44                        44                       44 

87 

7Ae"  steel 

2           " 

56,944 

94,910 

57,910 

76.3 

\ZJ9                     44                     44 

88 

7Ae"     " 

2           " 

59,130 

98,360 

61,130 

79-S 

44                      44                     44 

Double-welt  butt-joints;    J4-inch  iron  plates. 

615 

56"  iron 

i%  ins. 

53,475 

67,321 

16,944 

62.2 

1Me"  punched  holes. 

616 

56" 

i56 

50,959 

64,138 

16,719 

59-3 

456  RIVETED  JOINTS  AND  PIN  CONNECTION. 

TABLE  I. — Continued. 


[Ch.  IX. 


No. 

Size  of 
Rivet 
and 
Kind. 

Pitch  of 
Rivet. 

Maximum  Stresses, 
Pounds  per  Square  Inch. 

c 
'o 
'"*+> 

o'  ,r 
•W 

Remarks. 

Tension 
on  Net 
Area  of 
Plate  (  D 

Com- 
pression 
on  Dia- 
metral 
Surface 

Shearing 
on 
Rivets 
(S). 

Single-riveted  lap-joints;   %-inch  iron  plates. 


62 

63 

64 
65 

66 
67 

720 
721 

iVie"iron 

i' 
i'          " 

2      ins. 

2 

2 
2            " 

1%      " 

iH   " 

2^6  " 
2^16  " 

37,460 

36,130 
38,190 
36,210 
41,750 
41,290 
61,700 
58,510 

60,340 
58,150 

60,730 
57,530 
54,130 
.  53,400 
52,970 
50,220 

38,280 

35,520 
37,530 
36,050 

34,230 
34,150 
26,180 
24,830 

49.0 
47-2 
49-7 
47.1 
50  .  o 
49-3 
60.4 
57-i 

%/;  punched  holes. 
"      drilled       ;; 
"    punched  holes, 
i  He"    "t 

Single-riveted  lap-joints;    %-inch  steel  plates. 


51 

1:He"iron 

ins. 

39,220 

63,210 

39,740 

45-  4 

%"      punched  holes. 

52 

*        " 

** 

37,700 

60,760 

38,190 

43-6 

"               " 

53 

'    steel 

" 

55,215 

89,580 

56,430 

64.1 

"               "             ' 

54 

i       « 

M 

54,740 

88,660 

55,460 

63.5 

"               " 

55 

i       11 

M       ' 

63,650 

80,930 

50,650 

66.7 

drilled 

56 

'       " 

%       ' 

63,976 

81,600 

50,900 

67.2 

"               " 

238 

%'       " 

' 

65,460 

89,490 

53,560 

70.9 

1%6//  punched 

239 

H'      " 

' 

65,210 

88,990 

53,600 

70.  6 

"             " 

718 

i'  iron 

%6    ' 

73,394 

79,510 

36,614 

71  .4 

i  He"     ;; 

719 

i'      " 

%6  " 

73,970 

80,200 

36,590 

72.0 

Double-riveted  lap-joints;   %-inch  iron  plates. 


68  i#e"ii 

69 

58 

70 
71 
81 
82 


steel 


48,450 

39,160 

24,760 

63-5 

50,730 

41,070 

26,150 

66.4 

50,220 

40,640 

25,330 

65-7 

46,255 

41,480 

27,550 

60.5 

46,110 

41,270 

27,010 

60.4 

30,920 

58,700 

39,130 

50.4 

30,130 

57,34° 

38,410 

49.  i 

punched  holes. 


drilled 


Double-riveted  lap-joints;  %-inch  steel  plates. 


62,800 
64,720 
63,210 
54,930 

44,660 
43,650 


50,760 
52,450 
56,860 
49,530 
84,460 
83,000 


32,310 
32,930 
34,7io 
30,830 
52,750 
51,845 


73-2 
75-  2 
73-  2 
63.8 
64.4 
63.0 


"  punched  holes. 


drilled 


Reinforced  riveted  lap-joints;   %-inch  iron  plates.     (See  figure  next  page.) 


244 

%"  iron 

j  2  ins.  joint 
i  4    "    welt 

38,870 

59,080 

40,360 

67.6 

i^ie"  drilled  hole,  %"welt. 

245 

%"     " 

{!   ','. 

43,770 

56,640 

34,46o 

74-0 

IS^g"               "                       »                «               » 

296 

H"    " 

\\   " 

44,840 

57,910 

33,890 

75-7 

"    14"     " 

2Q7.H"     " 

\:  :: 

42,680 

55,350 

31,810 

71.9 

"          "          "        "       " 

Art.  76.]  TESTS   OF  FULL-SIZE  RIVETED   JOINTS. 

TABLE  I. — Continued. 


457 


Maximum  Stresses, 

. 

Pounds  per  Square  Inch. 

c 
•o 

. 

Size  of 

No. 

Rivet 
and 
Kind. 

Pitch  of 
Rivet. 

Tension 
on  Net 

Com- 
pression 
on  Dia- 

Shearing 
on 

o  § 

x9 

O  t_ 
C  0) 

Remarks. 

Area  of 

metral 

Rivets 

Plate  (T) 

Surface 

(S). 

£ 

(/). 

W 

L                                          I      ^ 

Reinforced  riveted  joints;     %-inch  steel  plates.     (See  above  figure.) 

246 

H"  steel  itjJ^SSt} 

62,050 

67,320 

32,960 

89.0 

i%e"  drilled  holes. 

247 

H"     "       \      '•'              } 

62,880 

68,135 

33,900 

90.  i 

,. 

298 

%"  iron 

i  "•     I 

61,020 

67,300 

34,250 

87.8 

• 

299 

W   " 

\  "    \ 

61,710 

68,040 

34,750 

88.9 

" 

Single-riveted  lap-joints;   J^-inch  iron  plates. 

240 
241 
292 
293 
327 
328 

%"  iron 

!!  st»el 

ins. 

31,100 
31,395 

32,376 
33,i8o 
39,900 
40,500 

41,500 
41,955 
47,-Sso 
48,890 
58,880 
59,900 

34,280 
34,960 
38,020 
39,220 
47,020 
47,830 

39-8 
39-7 
42.9 

44-3 

52.  2 

54-2 

lsAa"  punched  holes. 
))       drilled       •* 

Single-riveted  lap-joints;   J^-inch  steel  plates. 

242|?4"   iron 
243i;J4" 
294  s%«  ' 

295,1!>i6"   " 

ins. 

38,204 
35,9i5 
60,210 
49,590 

50,940 
47,890 
56,980 
47,060 

41,100 
38,636 
36,770 

30,540 

38.2 
35-9 
51  .  2 
42.2 

13/i(6"  punched  holes, 
i" 

Double-riveted  lap-joints;    J^-inch  iron  plates. 

329 
635 

•K"  iron    (2  ins. 

%"           "     \2      " 

44,320 
42,920 

59,640 

57,950 

25,280 
24,560 

57-0 
55-2 

i%e"  punched  holes. 

61 9l15A(i"iron|2  ins. 
6aoU%e"    "   I2    " 


7  30( 


732[i"iron     |2%ins. 
733U"    •'          U^    " 


Double-riveted  lap-joints;   ^-inch  steel  plates. 

I    64,602    I    29,354   I    19,670    I  53-8  li"  punched  holes. 
I    64,519    I    29,371    I    19,644   1  53.8)  " 

Single-riveted  lap-joints;    %-inch  iron  plates. 


34,680 
34,230 


47,510 
46,790 


35,460    I  44.9 
34,930    I  42.0 


"  punched  holes. 


Double-riveted  lap-joints;   %-inch  iron  plates. 

I   43,580    I    29,740    I    22,960    I  56.  3  I  iMs"  punched  holes. 
|    45,850    I    3i,3io    I    23,670    I  59-  3  I     ' 


Single -riveted  lap-joints;   %-inch  steel  plates. 

734(1"  steel   (2%  ins.  I   49,650   I   56,760   I   43,490   I  50.5  U#e"  punched  holes. 

735li"  h%    "  I    52,770    I    60,150    |    46,080    I  53-61     ' 

Double-riveted  lap-joints;    ^-inch  steel  plates. 

736[T"steel    h^^ins.  I    69,680    I    30,780    I    30,470    I  70. 9  I  i^e"  punched  holes. 

737lr"      '         12%    "  I    67,100    I    38,300    I    29,340    168.3)     ' 


458 


RIVETED  JOINTS  AND  PIN  CONNECTION. 


[Ch.  IX. 


TABLE  II. 
RIVETED  JOINTS—  IRON  AND  STEEL. 


Maximum  Stresses, 

. 

Pounds  per  Square  Inch. 

.5 

No. 

Thick- 
ness of 
Plate 
and 

Diameter 
and 
Kind  of 
Rivet. 

Pitch  of 
Rivet. 

Tension 
on  Net 

Com- 
pression 
on  Dia- 

Shearing 
on 

|| 

Remarks. 

Kind. 

Area  of 

metral 

Rivets 

4|04 

Plate  (J) 

,  Surface 

(5). 

1 

(f). 

W 

Single-riveted  iron  lap-joints. 


1Vie"  iron 


i%  ins. 

39,300 

50,850 

33,710 

"      " 

41,000 

53,050 

35,170 

2 

35,650 

47,350 

37,300 

" 

35,150 

46,690 

36,780 

Single-riveted  iron  butt-joints. 

2     ins. 

46,360 

72,390 

25,380 

**       ** 

46,875 

73,050 

25,450 

"       " 

46,400 

61,940 

24,630 

a       tt 

46,140 

61,740 

24,310 

2%      " 

44,260 

60,330 

23,010 

42,350 

58,080 

22,310 

2.9     " 

42,310 

57,000 

21,870 

41,920 

56,540 

22,140 

punched  holes. 


47.0 
49.0 

45-6 
44-9 


59-9  %"     punched  holes. 

60.5  ' 

59-4 

59-2 

57-2 

54-9 

52.i 

51-7 


Single  -riveted  steel  lap-joints. 


steel 


61,270 
60,830 
47,530 
49,840 


65,760 
65,320 
44,590 
46,960 


40,390 
39,900 
29,390 
31,070 


59-5 
59-1 
40.2 
42.3 


1%e"  punched  holes. 


Single-riveted  steel  butt-joints. 


17 
18 
19 

20 
21 
22 
23 
24 

%'J  steel 

2  '           " 

1Me"  iron 
!',     st»el 

2      ir 

2% 

S. 

62,770 
61,210 
68,920 
66,710 
62,180 
62,590 
54,650 
54,200 

97,940 
95,210 
62,220 
59,58o 
71,450 
71,930 
55,6io 
55,840 

31,240 
31,020 
20,370 
19,890 
27,750 
27,940 
23,190 
22,810 

71.7 
69.8 
57-1 
55-0 
63-4 
63.8 
54-0 
53-4 

%"      pun 
i 

ched  ho 

les. 

T  :: 

•*"  :: 

2# 

It  is  important  to  notice  that  in  general  the  highest  ulti- 
mate resistances  of  tension  and  compression  or  bearing  are 
found  with  the  thin  plates,  and  that  those  quantities 
diminish  appreciably  as  the  thickness  of  plate  increases, 
both  for  iron  and  steel.  This  law  is  not  so  well  defined  in 
reference  to  the  diameter  of  rivet,  if  indeed  these  tests  show 
it  at  all,  except  for  steel. 


Art.  76.]  TESTS   OF  FULL-SIZE   RIVETED   JOINTS.  459 

The  length  of  these  test  joints  varied  from  9.75  to  13 
inches  for  Tables  I  and  II,  and  from  10  to  27  inches  for 
Table  III. 

Although  the  results  of  these  tables  are  somewhat 
irregular,  they  confirm  the  general  accuracy  of  the  relations 
established  between  the  values  of  T,  /,  and  5  in  the  pre- 
ceding articles,  as  well  as  other  general  rules  and  conclu- 
sions for  boiler  work. 

Some  efficiencies  are  lower  than  those  given  for  similar 
joints  in  Art.  94,  but  such  instances  can,  by  the  aid  of  the 
tables,  be  traced  either  to  indifferent  design  or  a  phenome- 
nally low  value  of  some  one  of  the  three  resistances.  In 
general  the  results  compare  well  with  those  given  in  that 
article. 

The  pitches  of  rivets  are  seen  to  be  adapted  to  boiler 
work,  being  much  less  than  are  ordinarily  used  in  bridge 
work;  yet  the  corresponding  resistances  show  what  may 
legitimately  be  done  and  expected  when  unusual  condi- 
tions demand  a  departure  from  ordinary  rules. 

Before  deducing  working  intensities  for  bridge  con- 
struction from  the  preceding  results  it  is  to  be  first  ex- 
plained that  those  results  are  as  given  in  the  government 
reports,  and  that  the  net  section  used  is  the  gross  section 
of  the  plate,  less  the  actual  metal  removed  by  the  punch  or 
drill,  with  no  allowance  for  deterioration  by  the  former  in  the 
immediate  vicinity  of  the  hole.  Again,  in  Tables  I  and  II 
the  diametral  bearing  surface  and  the  shearing  area  of  the 
rivet  are  taken  to  be  those  of  the  drill,  or  a  mean  between 
the  punch  and  die  in  case  of  punched  holes.  In  bridge 
work,  in  determining  the  net  section,  metal  is  deducted 
for  a  diameter  equal  to  that  of  the  cold  rivet  before  driving 
plus  one  eighth  of  an  inch ;  and  the  shearing  and  bearing 
are  computed  for  the  section  and  diameter  of  the  cold  rivet 
before  driving. 


460  RIVETED  JOINTS  AND  PIN   CONNECTION.  [Ch.  IX. 

TABLE  III. 
TESTS  OF  STEEL-RIVETED  JOINTS;   HNCH  PLATES. 


Maximum  Stresses:  Pounds  per 

Square  Inch. 

Efficiency 

Joint. 

Rivet. 

Tension  on 

Compres- 

Shearing 

of  Joint, 
Per  Cent. 

Remarks. 

Net  Area 

sion  on 

on 

of  Plate 

Diametral 

Rivets 

(T). 

Surf  ace  (f). 

(5). 

A 

j"  steel 

38,940 

57,96o 

41,760 

47.1 

J-i"  drilled  holes. 

B 

39,450 

81,530 

35,560 

57 

it 

C 

62,200 

59,950 

22,480 

83.5 

56,410 

77,900 

29,640 

80.3 

' 

63,000 

8S,sio 

20,930 

85.5 

' 

59,330 

78,900 

29,410 

85-3 

* 

55,050 

71,890 

29.850 

79-4 

i 

5i,340 

76,550 

36,030 

78 

52,150 

50,170 

20,790 

78.6 

62,390 

54,660 

21,530 

90.  i 

58,550 

5i,35o 

20,620 

84.7 

55,030 

67,490 

27,030 

82.5 

*  Joint  not  fractured. 

A.  Double-riveted  lap-joint;  -fc-inch  plate. 

B.  Double-riveted  butt-joint,  two  splice-plates;  -^-in.  plate. 

C.  Treble-riveted  " 

H.  Quadruple-riveted  butt-joint,  two  splice-plates;  J-in.  plate. 

The  pitch  of  the  outside  rows  of  rivets  in  joints  B,  C,  and  H  was  double  that  of  the 
adjoining  rows.  In  the  same  joints  one  splice-plate  was  narrower  than  the  other,  so  that 
it  took  one  less  row  of  rivets  on  either  side  of  the  joint  than  the  other. 

With  these  explanations  in  view,  the  preceding  tests 
justify  the  following  working  stresses  for  the  plate-girder 
floor-beams  and  stringers  of  railway  bridges  with  machine- 
driven  rivets. 


!7,5°°  'bs.  per  sq.  in.  for  iron. 
10,000    •;      "     -     "     "     steel. 

(  14,000  Ibs.  per  sq.  in.  for  iron. 
Rivet  bearing {  ig)000    «      «    <«     .,     «    steel 

C    8,000  Ibs.  per  sq.  in.  for  iron. 
Tension  in  net  section  of  plate  {  T  0  000    «      «     « «    «    «    steej 


The  bearing  resistances  are  taken  rather  low,  especially 
for  steel,  for  the  reason  that  thick  plates  are  frequently 
used  in  bridge  construction,  and  the  ultimate  bearing 


Art.  76.  TESTS    OF  FULL-SIZE  RIVETED  JOINTS.  461 

resistance  for  them  is  appreciably  less  than  for  the  thin 
plates  used  in  most  of  the  preceding  tests. 

The  preceding  working  stresses  aie  based  on  steel  for 
rivets  giving  from  56,000  to  64,000  pounds  per  square  inch 
tensile  resistance,  while  the  steel  for  plates,  in  test  speci- 
mens, should  offer  from  58,000  to  66,000  pounds  per  square 
inch  ultimate  tensile  resistance. 

In  the  government  report  from  which  Table  I  is  ab- 
stracted, can  be  found  a  large  number  of  tests  made  for 
the  purpose  of  determining  the  proper  minimum  distance 
from  the  centres  of  rivet-holes  to  the  edge  of  plates.  As 
a  result  of  those  tests  and  other  experience  on  the  same 
subject,  it  may  be  stated  that  the  least  distance  from  the 
centre  of  a  rivet-hole  to  the  edge  of  a  plate  may  be  taken 
at  one  and  one  half  the  diameter  of  the  hole  for .  steel  and 
one  and  five  eighths  the  diameter  of  the  hole  for  iron,  in 
cases  where  it  is  important  to  secure  the  maximum  resist- 
ance of  the  joint. 

Efficiencies. 

The  values  of  the  quantity  which  has  been  termed  the 
•'  efficiency  "  of  the  joint,  i.e.,  the  ratio  of  the  resistance  of  a 
given  width  of  joint  over  that  of  an  equal  width  of  solid 
plate,  in  the  preceding  investigations,  are  those  actually 
determined  by  experiments  with  the  joints  themselves. 
They  may,  therefore,  be  relied  upon.  Some  values  which 
have  for  many  years  been  considered  as  standard,  but 
which  in  reality  are  of  a  somewhat  arbitrary  nature,  and 
at  best  belonging  to  a  limited  class  of  joints,  have  been 
disregarded. 


462  RIVETED  JOINTS  AND  PIN   CONNECTION.  [Ch.  IX. 

The  tests  of  full-size  wrought-iron  and  steel-riveted 
joints  exhibited  in  Art.  76  show,  as  a  rule,  that  thin  plates 
give  materially  higher  efficiencies  than  thick  plates.  Al- 
though there  are  irregularities,  single-riveted  lap-joints  may 
yield  efficiencies  running  from  50  to  74  per  cent,  for  |-inch 
plates,  but  dropping  to  50  to  54  per  cent,  for  f-inch  plates 
and  materially  lower  for  ^-inch  plates.  On  the  whole, 
the  double-riveted  lap-joints  show  somewhat  higher  effi- 
ciencies than  the  single-riveted,  but  not  quite  the  same 
relative  differences  between  J-inch  and  f-inch  plates,  the 
values  being  found  more  generally  between  about  60  and 
80  per  cent. 

The  single-riveted  butt-joints  of  Table  II,  Art.  76, 
give  efficiencies  ranging  from  about  52  to  72  per  cent. 

Some  unusually  high  efficiencies  are  found  in  Table  III 
of  the  same  article  for  butt-joints,  i.e.,  about  78  to  90  per 
cent.  Those  high  values  are  due  to  the  special  design  of 
the  joints,  and  they  cannot  ordinarily  be  attained  in  prac- 
tice, but  they  show  that  well-considered  designs  will  yield 
greatly  increased  efficiencies. 

In  general,  efficiencies  running  from  65  to  70  per  cent, 
may  be  considered  excellent  for  the  usual  conditions  of 
practice. 

Art.  77. — Tests  of  Joints  for  the  American  Railway  Engineering 
and  Maintenance  of  Way  Association  and  for  the  Board  of 
Consulting  Engineers  of  the  Quebec  Bridge. 

In  "  Proceedings  of  the  American  Railway  Engineering 
and  Maintenance  of  Way  Association,"  Vol.  6,  1905,  there 
are  given  the  results  of  a  series  of  tests  of  carbon-steel  riveted 
joints  and  a  duplication  of  that  series  of  tests  in  both  nickel 
and  chrome-nickel  steel  made  for  the  Board  of  Consulting 
Engineers  of  the  Quebec  Bridge  by  Profs.  Arthur  N.  Talbot 


Art.  77-} 


TESTS   OF  JOINTS. 


463 


and  Herbert  F.  Moore  of  the  University  of  Illinois,  also  fully 
described  in  Bulletin  No.  49  (1911)  of  that  institution. 
There  were  144  joints  tested  in  the  latter  two  series. 
Furthermore,  there  were  tested  in  alternate  tension  and 
compression  16  other  nickel-steel  joints  and  the  same 
number  of  chrome-nickel  steel  joints. 

All  the  main  plates  of  these  joints  were  6.5  inches  or 
7.5  inches  wide  with  thicknesses  from  f  inch  to  f  inch 
except  the  32  joints  subjected  to  compression,  for  which 
the  plates  were  2  inches  thick.  There  were  24  lap  joints, 
the  same  number  of  butt-joints  with  double  covers  or  butt- 
straps  and  an  equal  number  each  of  the  same  type  of 
joint  with  one  filler  and  two  fillers  on  each  side  of  both 
main  plates.  The  remaining  joints  for  tension  loads  only 
(yixf-inch  main  plates),  with  the  exception  of  two  sets  of 
eight  each,  were  also  made  with  one  or  two  fillers,  but  the 
latter  extended  beyond  the  end  of  the  cover  far  enough  to 
take  one  rivet. 

All  rivets  were  f -inch  in  diameter,  and  those  driven  by 
a  hydro-pneumatic  riveter  were  called  "  shop  "  rivets  while 
those  driven  by  a  hand-pneumatic  riveter  were  designated 

TABLE   I. 

CHEMICAL   COMPOSITION    OF   RIVET  AND   PLATE   MATERIAL 


Element. 

Nickel-steel 
Riveted  Joints. 

Chrome-nickel-steel 
Riveted  Joints. 

Rivet 
Material 
Per  Cent. 

Plate 
Material 
Per  Cent. 

Rivet 
Material 
Per  Cent. 

Plate 
Material 
Per  Cent. 

Carbon  

o.  141 
o  .  0023 

0.037 
0.442 
3-33 

0.258 
0.008 
0.044 
0.700 
3-330 

o.  136  ' 

0.038 
0.032 
0.696 
0.986 

o  .  240 

O.I9I 
0-035 
0.042 
0.485 

0-733 
0.170 

Sulphur 

Phosphorus 

Manganese 

Nickel 

Chromium    

464 


RIVETED  JOINTS  AND   PIN  CONNECTION. 


[Ch.  IX. 


TABLE   II. 

PHYSICAL   PROPERTIES   OF   RIVET   AND   PLATE   MATERIAL 

All  stresses  in  pounds  per  square  inch. 


Item. 

Nickel-Steel.    , 

Chrome-Nickel-Steel. 

Rivet 
Material. 

Plate 

Material. 

Rivet 
Material. 

Plate 
Material. 

Number  of  specimens 
tested  

2 

9 
40,200 
5i,7oo 
89,700 

25.0 

55-8 
29,950,000 

2 
38,400 

59,000 
35-2 

63.3 

8 
27,200 
36,300 
63,900 

31-7 

59-9 
30,750,000 

Elastic  limit 

Stress  at  yield  point  .  .  . 
Stress  at  ultimate  .... 
Elongation  in  2  inches, 
per  cent  

45,000 
68,500 

33-5 
63.4 

Reduction  of  area,  per 
cent  
Modulus  of  elasticity 

as  "field  "  rivets.  The  difference  in  resistance  of  the  shop 
and  field  rivets  was  not  material. 

Tables  I  and  II  show  the  chemical  composition  and  the 
physical  properties  of  the  nickel  and  chrome-nickel  steels 
used. 

The  following  statement  shows  in  a  condensed  form 
the  results  of  the  tests. 


TABLE    III. 

Nickel-Steel  Joints 

Av.  Ult.  shear  shop  and  field  rivets 

Max.  tension  in  plates 


Lbs.  per  Sq.  In. 
52,440    to    60,140 
16,850    to    50,800 


Chrome- Nickel- Steel  Joints 

Lbs.  per  Sq.  In. 

Av.  Ult.  shear  in  rivets 48,190     to    56,650 

Max  tension  in  plates 16,170     to    49,500 

Carbon  Steel  (Main,  of  Way  Assoc.  Tests) 

Lbs.  per  Sq.  In. 

Av.  shear  stress 44,940    to    52,060 

Max.  tension  in  plates 15,190    to    48,400 


Art.  77.]  TESTS  OF  JOINTS.  465 

The  shearing  of  the  rivets  caused  the  failures  of  all  the 
nickel-steel  and  chrome-nickel  steel  joints. 

The  "  carbon  steel  "  used  in  the  American  Railway  Engi- 
neering and  Maintenance  of  Way  Association  tests  was  low 
basic  open  hearth  material  conforming  to  the  specifications 
of  that  Association.  Some  of  these  joints  failed  by  the 
yielding  of  the  plates  but  the  greater  part  of  them  failed 
by  the  shearing  of  the  rivets  and  the  results  are  all  given 
in  terms  of  the  maximum  shearing  stress  in  the  rivets  at 
the  instant  of  failure. 

The  lower  values  in  the  ultimate  and  final  shear 
stresses  in  these  series  of  tests  belong  to  the  longer  rivets, 
i.e.  to  the  joints  in  which  fillers  were  used.  This  was  to 
be  expected  in  consequence  of  the  increased  bending  in 
those  rivets.  Indeed,  these  tests  indicate  that  with  ordi- 
nary thicknesses  of  plates  the  carrying  capacity  of  the 
rivets  begins  to  be  seriously  affected  when  the  "  grip  "  of 
the  rivets,  i.e.  the  aggregate  thickness  of  piates  pierced  by 
them,  exceeds  about  four  diameters.  It  should  be  stated, 
however,  that  this  depends  much  upon  the  design  of  the 
joint. 

Friction  of  Riveted  Joints. 

Careful  observations  were  made  by  Profs.  Talbot  and 
Moore  as  well  as  in  the  tests  of  joints  for  the  American 
Railway  Engineering  and  Maintenance  of  Way  Association 
to  determine  the  friction  of  riveted  joints  which  experienced 
engineers  have  long  known  to  exist.  These  observations 
indicate  that  a  material  slipping  of  the  plates  took  place  in 
some  of  these  joints  when  the  shearing  stress  in  the  rivets 
was  not  greater  than  about  6,000  pounds  per  square  inch. 
In  other  cases  this  slipping  took  place  when  the  rivet  shear 
was  as  high  as  15,000  pounds  per  square  inch.  It  was 
observed,  as  might  be  anticipated,  that  the  quality  of  the 


466  RIVETED  JOINTS  AND  PIN  CONNECTION.  [Ch.  IX. 

material  of  the  joints  had  little  effect  upon  the  degree  of 
stress  at  which  slipping  began.  The  results  were  about 
the  same  for  the  low  carbon  steel  joints  as  for  the  chrome- 
nickel  steel  joints.  As  might  be  expected  in  a  well-pro- 
portioned joint,  the  friction  between  the  plates  depends 
upon  the  force  with  which  they  are  held  in  contact  by  the 
rivets.  The  motion  of  the  plates  is  obviously  due  to  the 
fact  that  the  shaft  of  the  rivet  in  cooling  contracts  more 
than  the  comparatively  cool  plate  around  it  leaving  a  small 
annular  space  between  the  rivet  and  the  wall  of  the  hole. 
As  the  load  on  the  joint  increases  a  degree  of  direct  stress 
of  tension  (or  of  compression  in  joints  under  compression) 
is  reached  at  which  the  plates  slip  on  each  other  bringing 
the  rivet  shafts  successively,  or  more  or  less  simultaneously, 
in  contact  with  the  bearing  side  of  the  hole. 

After  the  load  increases  still  more,  a  higher  stage  of 
stress  is  reached  at  which  the  yield  point  of  the  joint  is 
found  when  relatively  rapid  distortion  takes  place.  As  an 
average  the  yield  point  of  the  nickel  steel  joints  was  found 
at  an  intensity  of  shearing  stress  in  the  rivets  of  about 
35,000  pounds  per  square  inch  and  not  much  different  from 
that  for  the  chrome-nickel  steel  joints.  Material  bending 
of  the  rivets  appears  to  be  an  influential  element  in  the 
increased  deformation  at  the  yield  point  of  a  joint  and  it 
is  reasonable  to  suppose  that,  other  things  being  the  same, 
the  longer  the  rivets  the  lower  will  be  the  degree  of  stress 
at  which  the  yield  point  is  found,  although  it  is  doubtful 
whether  the  rivets  are  long  enough  in  the  well-designed 
riveted  joints  of  good  engineering  practice  to  show  much 
effect  upon  the  yield  point.  Profs.  Talbot  and  Moore 
state  that  "  The  ratio  of  the  yield  point  of  riveted  joints 
to  ultimate  shearing  strength  in  these  tests  was  about  the 
same  as  the  ratio  of  the  yield  point  of  the  plate  material  in 
tension  to  the  ultimate  tensile  strength  of  the  plate  material." 


Art.  78.]  RIVETED-TRVSS  JOINTS.  467 

The  results  obtained  from  the  joints  tested  in  alternate 
tension  and  compression  were  not  markedly  different  from 
those  obtained  in  tension.  The  yield  point  seems  to  be 
slightly  lowered  after  a  few  alternations  of  tensile  and  com- 
pressive  loads.  If  these  alternations  took  place  rapidly, 
doubtless  the  joints  would  show  much  diminution  of  re- 
sisting capacity  but  the  actual  alternations  were  few  in 
number  and  not  rapidly  made. 

These  tests  show  that  the  friction  between  plates  of  a 
riveted  joint  cannot  properly  be  considered  as  enhancing 
the  resisting  capacity.  Furthermore,  this  slipping  has  a 
direct  bearing  upon  the  computations  of  secondary  stresses 
in  trusses  with  riveted  connections.  The  corresponding 
deformation  may  militate  materially  against  the  accuracy 
or  reliability  of  such  computations. 

Art.  78.— Riveted-truss  Joints. 

The  circumstances  in  which  riveted  joints  are  used  in 
truss  work  render  permissible  many  special  forms  which 


OO 

oo 


FIG.  i. 

can  find  no  place  in  boiler-riveting.  If  joints  are  found 
under  the  same  circumstances,  as  far  as  the  transference 
of  stress  is  concerned,  precisely  the  same  forms  would  be 
used,  except  that  calking  is,  of  course,  only  required  in 
boiler  work. 


F     G 


468  RIVETED  JOINTS  AND  PIN  CONNECTION.  [Ch.  IX. 

Fig.  i  shows  a  common  form  of  chord  construction  in 
riveted-truss  work, with  the  relative  proportions  exaggerated. 

The  lower  portion  of  the  figure  shows         M  N 

a  section  of  the  chord  in  which  the  cover- 
or  splice-plate  is  shaded.  The  joint  is 
supposed  to  be  in  tension.  v^^^ 

In  this   form  of  joint  the  splice-pla*te  d 

material  is  reduced  to  a  minimum.     These  (j 

are,  in  reality,  two  lap-joints  CD  and  DE 
with  the  two  plates  C  and  E  to  be  spliced.  C 

In  each  lap-joint  there  should  be  sufficient 
rivets  determined  by  the  methods  of  Art.  74.  The  splice- 
plate  A  B  should  be  long  enough  to  give  the  requisite  plate 
AC  to  the  left  of  C,  with  the  same  length  from  B  to  a  point 
vertically  over  E. 

In  most  cases  one  or  two  plates  only  should  be  spliced 
at  the  same  point. 

The  joint  in  the  vertical  plate  should  be  formed  as  at 
FG\  i.e.,  it  should  be  a  double-cover  butt-joint.  The 
principles  already  established  in  a  preceding  section,  in 
regard  to  the  thickness  of  covers  and  diameter  of  rivets, 
should  be  observed  here. 

The  two  or  more  full  rows  of  rivets  on  either  side  of  the 
joint  may  as  well  be  chain-riveted  with  a  pitch  of  3 J  to  4 
diameters.  Other  rivets  should  then  be  staggered  in  until 
the  group  of  rivet  centres  on  each  side  is  brought  to  a  point, 
as  shown  in  the  upper  part  of  Fig.  i.  In  this  manner  the 
available  section  of  a  width  of  plate  equal  to  that  of  the 
cover  becomes  approximately  equal  to  the  total,  less  the 
material  from  one  rivet-hole.  Hence  the  efficiency  of  the 
joint  becomes  correspondingly  increased. 

If  the  joint  is  in  compression  the  preceding  observa- 
tions hold  without  change,  except  that  all  covers  should 
have  the  same  thickness  as  the  plates  covered. 


Art.  78.]  RIVETED-TRVSS  JOINTS.  469 

Even  if  the  joints  C,  D,  E,  and  H  are  of  planed  edges, 
little  or  no  reliance  should  be  placed  upon  their  bearing  on 
each  other,  since  the  operation  of  riveting  will  draw  them 
apart  more  or  less,  however  well  the  work  may  be  done. 

Unless  great  caution  is  observed  and  excellence  of  design 
secured,  there  will  frequently  be  excessive  bending  in  the 
riveted  joints  of  truss  work,  on  account  of  the  great  variety 
of  connections  required. 

Diagonal  Joints. 

Diagonal  riveted  joints  have  from  time  to  time  been 
proposed,  the  line  of  the  joint  making  an  angle  of  perhaps 
45°  to  the  line  of  action  of  the  loading.  Such  joints 
when  properly  designed  have  high  efficiencies  for  the  reason 
that  a  normal  section  of  the  joint  taken  anywhere  within 
its  extreme  limits  will  lie  wholly  within  the  main  plates 
except  at  the  point  where  the  oblique  joint  line  cuts  it.  In 
designing  such  a  joint,  however,  the  rows  of  rivets  should  be 
placed  parallel  to  the  joint  line  and  extend  across  the  entire 
main  plates,  or  some  other  arrangement  may  be  employed 
which  will  make  the  centre  of  gravity  of  the  group  of  rivets 
on  the  two  sides  of  the  joint  lie  in  the  centre  line  of  the 
main  plates  or  other  connected  members.  If  this  con-' 
dition  is  not  attained,  there  will  be  eccentricity  of  the 
aggregate  resistance  of  the  rivets  on  either  side  of  the  joint 
line  resulting  in  serious  bending  about  an  axis  perpendicular 
to  the  main  plates.  The  added  cost  of  this  type  of  joint 
and  the  inconvenience  of  its  use  in  many  cases  prohibits 
its  general  employment  as  a  detail  in  riveted  structural 
work. 

Riveted  Joints  in  Angles. 

It  has  been  found  by  tests  of  full-size  angles  that  if  a 
riveted  joint  be  formed  by  riveting  one  leg  only,  the  ulti- 


470  RIVETED  JOINTS  AND  PIN  CONNECTION.  [Ch.  IX. 

mate  tensile  resistance  per  square  inch  of  the  net  angle 
section  may  be  but  75  per  cent,  of  the  ultimate  tensile 
resistance  of  test  specimens  cut  from  the  same  angle.  On 
the  other  hand,  if  both  legs  are  riveted  the  ultimate,  tensile 
resistance  per  square  inch  of  the  net  section  may  easily 
be  90  per  cent,  of  the  ultimate  resistance  of  test  specimens 
cut  from  the  same  angle.  These  results  show  that  both 
legs  of  angles  should  always  be  riveted  at  joints. 

Hand  and  Machine  Riveting. 

The  development  of  the  pneumatic  and  other  power 
riveters  for  both  shop  and  field  purposes  has  practically 
eliminated  hand  riveting  from  all  structural  work  except 
in  rare  cases.  When  hand  riveting  was  done  its  inferiority 
to  power  riveting  was  recognized  by  specifying  that  at 
least  one-third  more  rivets  should  be  used  when  they  were 
driven  by  hand. 

Art.  79.— Welded  Joints. 

Welded  joints,  as  a  rule,  have  never  been  permitted  in 
first  class  structural  work.  Fairly  good  joints  of  that  type, 
however,  were  made  where  necessary  in  wrought  iron,  but 
it  is  difficult,  if  not  essentially  impossible,  to  make  a  satisfac- 
tory weld  in  structural  steel  by  ordinary  procedure.  In  cases 
where  welding  of  steel  is  done,  some  method  is  necessary 
in  which  the  metal  at  the  weld  is  brought  into  a  state  of 
fusion  for  a  material  depth.  The  thermit  and  other  proc- 
esses accomplish  satisfactory  welded  joints  in  both  steel 
castings  and  rolled  bars  for  many  purposes  although  they 
are  not  used  in  structural  work. 

Art.  80. — Pin  Connections. 

A  pin  connection  consists  of  two  sets  of  eye-bars  or  links, 
through  the  heads  at  one  end  of  each  of  which  a  single  pin 


Art.  80.] 


PIN  CONNECTIONS. 


47i 


passes.     Fig.   i  shows  a  pin  connection;    A,  A,  B,  B\  are 
eye-bars  or  links,  and  P  is  the  pin. 


FIG.  i. 

The  head  of  the  eye-bar  (one  is  shown  in  elevation  in 
Fig.  2)  requires  the  greatest  care  in  its  formation.  It  is 
imperfect  unless  it  be  so  proportioned  that  when  the  eye-bar 
is  tested  to  failure,  fracture  will  be  as  likely  to  take  place 
in  the  body  of  the  bar  as  in  the  head ;  in  other  words,  unless 
its  efficiency  is  unity. 

In  Fig.  2  the  head  of  the  eye-bar,  or  link,  is  supposed 


FIG.  2. 


to  be  of  the  same  thickness  as  that  of  the  body  of  the  bar 
whose  width  is  w. 

If  t  is  the  thickness  of  the  bar  so  that  wt  is  the  area  of 
its  normal  section,  then  t  is  generally  included  between  the 


472  RIVETED  JOINTS  AND  PIN  CONNECTION.  [Ch.  IX. 

limits  of  \w  and  \w  for  ordinary  sizes  of  eye-bars.  These 
limits,  however,  are  exceeded  both  for  the  smaller  sizes  used 
and  the  larger  sizes.  A  bar  for  which  w=$  inches,  may 
have  a  thickness, of  if  inches,  while  the  maximum  thickness 
of  a  bar  16  inches  wide  may  be  no  'more  than  2  inches. 
Similarly  the  minimum  thickness  of  a  3 -inch  wide  bar  may 
be  |  inch  while  the  least  thickness  of  a  1 6-inch  wide  bar 
may  be  taken  at  i  f  inches  or  t  =  %w. 

In  the  early  days  of  eye-bar  manufacture  earnest  efforts 
were  made  to  analyze  the  complicated  condition  of  stresses 
in  the  eye-bar  head  so  as  to  give  it  a  rational  outline,  and 
an  approximate  treatment  of  the  problem  may  be  found  in 
the  "  Trans.  Am.  Soc.  of  Civ.  Engrs."  Vol.  VI,  1877,  the 
results  of  which  agree  essentially  with  those  of  experi- 
ment. 

After  much  experimenting,  including  the  thickening  of 
the  head,  it  has  for  many  years  been  the  practice  to  make 
the  heads  of  eye-bars  circular  in  outline  as  shown  in  Figs. 
i  and  2 .  In  Fig.  2  the  front  part  of  the  head  NBM  is  a 
semicircle  and  it  is  extended  on  both  sides  to  the  left  of 
NM  so  as  to  be  tangent  to  the  circular  curves  of  the  neck 
drawn  with  the  radius  equal  to  the  width  d  of  the  entire 
head.  The  latter  curves  are  also  tangent  to  the  body  of  the 
bar  as  shown  at  H. 

The  head  is  formed  by  heating  the  end  of 'the  bar  to  a 
white  heat,  then  upsetting  it  in  a  properly-formed  die  as 
closely  as  possible  to  the  finished  shape.  A  little  finishing 
work  is  then  usually  done  under  a  power  hammer  or  between 
rolls.  The  head  is  seldom  thicker  than  the  body  of  the  bar. 

The  normal  section  of  the  head  taken  through  the  centre 
of  the  pinhole  is  usually  from  35  per  cent,  to  40  per  cent, 
in  excess  of  the  section  of  the  bar.  All  steel  eye-bars  are 
thoroughly  annealed  after  the  completion  of  manufacture 
so  as  to  remove  all  internal  stresses  in  the  head  and  any 


Art.  80.]  PIN  CONNECTIONS.  473 

undue  hardness  that  may  have  been  acquired  during  that 
process. 

The  diameter  of  the  pin  should  never  be  less  than  about 
80  per  cent,  of  the  width  w  of  the-  bar,  and  it  may  be  from 
if  to  i\  times  that  width,  the  greater  of  those  factors  be- 
longing to  bars  of  small  width  and  the  smaller  to  bars  of 
the  greatest  width  used. 

In  pin  connections  the  pin  is  subjected  to  heavy  bend- 
ing for  which  it  is  carefully  designed  as  well  as  for  the  shear 
in  its  normal  section  and  for  the  bearing  or  compression 
between  it  and  the  pin  hole.  The  pin  and  the  pin  hole 
are  accurately  machine  finished,  the  diameter  of  the  latter 
being  from  perhaps  T^7  inch  (for  the  smallest  pins)  to  -^ 
inch  (for  the  largest  pins)  greater  than  the  former. 

If  M  is  the  bending  moment  to  which  the  pin  is  sub- 
jected, k  the  greatest  intensity  of  bending  stress  devel-. 
oped,  and  A  the  area  of  the  normal  section  of  the  pin,  eq. 
(4)  of  Art.  90  gives 

M  =k  4^  =o.ikd*  (nearly),     .     .     .     .     (i) 

o 

or 

(2) 


Values  of  k,   for   circular   sections,   may  be   found   in 
Art.  QO. 


CHAPTER  X. 
LONG  COLUMNS. 

Art.  81. — Preliminary  Matter. 

THERE  is  a  class  of  members  in  structures  subjected  to 
compressive  stress  which  do  not  fail  entirely  by  compression. 
The  axes  of  these  pieces  coincide,  as  nearly  as  possible,  with 
the  line  of  action  of  the  resultant  of  the  external  forces, 
yet  their  lengths  are  so  great  compared  with  their  lateral 
dimensions  that  they  deflect  laterally,  and  failure  finally 
takes  place  by  combined  compression  and  bending.  Such 
pieces  are  called  "  long  columns,"  and  the  application  to 
them,  of  the  common  theory  of  flexure,  has  been  made  in 
Art.  35. 

Two  different  formulae  were  first  established  for  use  in 
estimating  the  resistance  of  long  columns ;  they  are  known 
as  "Gordon's  Formula"  and  "Hodgkinson's  Formula/' 
Neither  Gordon  nor  Hodgkinson,  however,  gave  the  original 
demonstration  of  either  formula. 

The  form  known  as  Gordon's  formula  was  originally  dem- 
onstrated and  established  by  Thomas  Tredgold  (''Strength 
of  Cast  Iron  and  other  Metals,"  etc.),  for  rectangular  and 
round  columns,  while  that  known  as  Hodgkinson's  formula 
(demonstrated  in  Art.  35)  was  first  given  by  Euler. 

In  1840,  however,  Eaton  Hodgkinson,  F.R.S.,  published 
the  results  of  some  most  valuable  experiments  made  by 

474 


Art.  81.] 


PRELIMINARY  MATTER. 


475 


himself  on  cast  and  wrought-iron  columns  (Experimental 
Researches  on  the  Strength  of  Pillars  of  Cast  Iron,  and 
other  Materials;  Phil.  Trans,  of  the  Royal  Society,  Part  II, 
1840),  and  from  these  experiments  he  determined  empirical 
coefficients  applicable  to  Euler's  formula,  on  which  account 
it  has  since  been  called  Hodgkinson's  formula. 

Prof.  Lewis  Gordon  deduced  from  the  same  experiments 
some  empirical  coefficients  for  Tredgold's  formula,  since 
which  time  Gordon's  formula  has  been  known. 

The  latter  has  been  quite  generally  used,  but  it  has 
lately  been  largely  displaced  by  the  straight-line  formula 
to  be  given  later.  Hodgkinson's  coefficients  and  formula 
have  now  been  abandoned. 

Before  taking  up  the  subject  of  long  columns  it  is 
desirable  to  establish  some  important"  properties  of  the 
moments  of  inertia  of  surfaces  used  in  the  analytic  treat- 
ment of  long  columns  and  in  some  problems  of  flexure. 

It  will  also  be  both  convenient  and  important  to  de- 
termine the  conditions  which  ex- 
ist with  an  isotropic  character  of 
section  in  respect  to  the  moment 
of  inertia. 

In  Fig.  i  let  BC  be  any  figure 
whose  area  is  A,  and  whose  cen- 
tre of  gravity  is  at  0.  In  the 
plane  of  that  figure  let  any  arbi- 
trary  system  of  rectangular  co- 
ordinates X'  ',  y  be  chosen  and  let  XY  be  any  other 
system  having  the  same  origin  ;  also,  let  #',  yf  and  x,  y  be 
the  coordinates  of  the  element  D  of  the  surface  A  in  the 
two  systems.  There  will  then  result 


x=xf  cos  a+y  sin  a, 
y=y*  cos  a  —  %'  sin  a. 


476  LONG  COLUMNS.  [Ch.  X. 

The  moments  of  inertia  of  the  surface  about  the  axes  y  and 
x  will  then  be 

1  x2dA  =  cos2  a  \  oc'2dA  +  2  sin  a  cos  a  (  x'y'dA  + 

sin2ajy2<M,      .     .     .     .     (i) 
J  y2dA  =  cos2  oj  /2<M  —  2  sin  a  cos  a(  x'y'dA  + 

sin2ajV2<M  .....     (2) 

If  A;  and  y  are  to  be  so  chosen  that  they  are  principal 
axes,  then  must  JxydA  =o,  or 

o  =  JxydA  =sin  a  cos  afy'^dA  +  (cos2  a  —  sin2  a)fx'yrdA 

—  sin  a:  cos  ajxf2dA  ;       (3) 


.'.  tan  2  a 


. 

fx'2dA-fy'2dA 


Hence,  since  tan  2  a  =  tan  (  180  +  2  a),  there  will  always 
be  two  principal  axes  90°  apart. 

Now,  if  f  x'y'dA  =o,  while  no  other  condition  is  imposed. 

tan  2a=-o.      This  makes  a=o  or  90°;    i.e.,  X'Yf  are  the 
principal  axes. 

If,  however,   (x'y'dA  =o,  while  a  is  neither  o  nor  90°, 
eq.  (3)  becomes 


or 


o 

tan  2a  =-,  i.e.,  indeterminate. 
o 


Art.  8i.  PRELIMINARY  MATTER.  477 

This  shows  that  any  axis  is  a  principal  axis ;  also  that 
fx'dA  =  fy2dA  =  fx'2dA  =  fy'2dA. 

Hence  the  surface  is  completely  isotropic  in  reference  to 
its  moment  of  inertia,  or  its  moment  of  inertia  is  the  same 
about  every  axis  lying  in  it  and  passing  through  its  centre  oj 
gravity. 

It  has  been  seen  that  this  condition  exists  where  there 
are  two  different  rectangular  systems,  for  which 

fxydA  =  fyfydA  =  o ; 

but  the  first  of  these  holds  true  if  either  x  or  y  is  an  axis  of 
symmetry,  and  the  latter  if  either  x'  or  y'  is  an  axis  of  sym- 
metry. 

Hence,  if  the  surface  has  two  axes  of  symmetry  not  at  right 
angles  -to  each  other,  its  moment  of  inertia  is  the  same  about  all 
axes  passing  through  its  centre  of  gravity  and  lying  in  it. 

Eqs.  (3)  and  the  two  preceding  it  also  show  that  the 
same  condition  obtains  if  the  moments  of  inertia  about  four 
axes  at  right  angles  to  each  other,  in  pairs,  are  equal. 

In  the  case  of  such  a  surface,  therefore,  it  will  only  be 
necessary  to  compute  the  moment  of  inertia  about  such  an 
axis  as  will  make  the  simplest  operation. 

Principal  Moments  of  Inertia. 

If  the  moments  of  inertia  I'  about  the  axis  of  Y'  and 
I"  about  the  axis  of  X'  be  expressed  in  terms  of  the  prin- 
cipal moments  /i  about  the  axis  of  Y  and  I2  about  the 
axis  of  X,  eqs.  (i)  and  (2)  will  give  by  simply  changing  the 


478  LONG  COLUMNS.  [Ch.  X. 

primes  from  the  second  to  the  first  members  of  the  equa- 
tions ; 

=  /'  =Ii  COS  2a+I2  Sin2  a.  .      .      .       (4) 


fy'2dA 


1"  =I2  cos2  a+Ii  sin2  a.  .     .     .     (5) 


If  the  principal  moments  of  inertia  I\  and  1  2  are  known 
eqs.  (4)  and  (5)  show  that  the  moments  T  and  I"  about 
any  axes  making  the  angle  a  with  the  principal  axes  may 
at  once  be  computed. 

Adding  eqs.  (4)  and  (5)  ; 

r+I"=Ii+l2=I  (Polar  moment)..     .     .     (6) 

Hence  the  sum  of  the  two  moments  of  inertia  about  any 
two  axes  at  right  angles  to  each  other  is  constant  and  equal 
to  the  polar  moment  of  inertia. 

If  the  second  members  of  eqs.  (4)  and  (5)  be  divided  by 
the  area  A  of  the  cross  section,  and  if  the  radii  of  gyration 
be  represented  by  r'  ,  r"  ,  r\  and  r2; 

r'2  =ri2  cos2  «+r22  sin2  a.  .     .     .    ...  .     (7) 

in2  a.  .     %  \  .  ~  :,-  '  .     (8) 


Each  of  eqs.  (7)  and  (8)  is  the  equation  of  an  ellipse  in 
which  ri  is  the  semi-axis  in  the  direction  of  the  coordinate 
axis  X  and  r2  is  the  semi-axis  of  the  ellipse  in  the  direction 
of  the  coordinate  axis  Y,  while  r'  and  r  "are  two  semi- 
diameters  ODr  and  OD,  all  as  shown  in  Fig.  2. 

If  eqs.  (7)  and  (8)  be  added,  eq.  (9)  will  result; 

.    ;..       .        .        .         (9) 


This  equation  is  the  expression  of  one  characteristic  of 
the  ellipse,  viz.,  the  sum  of  the  squares  of  any  two  conjugate 


Art.  81.] 


PRELIMINARY  MATTER. 


479 


semi-diameters  is  equal  to  the  sum  of  the  squares  of  the 
two  semi-axes.  The  two  radii  of  gyration  therefore  about 
any  two  inertia  axes  at  right  angles  to  each  other,  except 


FIG.  2. 


the  principal  axes,   are    semi-conjugate  diameters  of  the 
ellipse. 

Eqs.  (7)  and  (8)  are  precisely  the  same  in  character  as 
eq.  (3)  of  Art.  9  and  the  ellipse  of  Fig.  2  is  constructed  pre- 
cisely as  was  the  ellipse  of  stress.  The  two  principal  radii 
of  gyration  r\  and  r2  are  represented  by  the  semi-axes  OA 
and  OB,  while  the  semi-conjugate  diameters  OD'  and  OD 
represent  the  radii  of  gyration  r'  and  r"  taken  about  any 
two  axes  at  right  angles  to  each  other,  represented  by  ON 
and  ON'.  The  construction  lines  of  Fig.  2  show  how  the 


480  LONG  COLUMNS.  [Ch.  X. 

ellipse  is  constructed  from  eqs.   (7)   and  (8),  precisely  as 
was  the  ellipse  of  stress  in  Art.  9. 

If  it  is  desired  to  find  the  radius  of  gyration  about  any 
axis,  as  the  semi-diameter  OQ,  the  construction  of  the 
ellipse  shows  that  it  is  only  necessary  to  describe  the  two 
circles  with  radii  r\  and  r^  as  shown  in  the  figure,  then 
erect  ON  perpendicular  to  OQ  and  draw  the  horizontal  and 
vertical  lines  respectively  from  N  and  K  to  their  intersection 
D  on  the  ellipse.  The  semi-diameter  OD  will  be  the  radius 
of  gyration  desired  and  its  direction  on  the  figure  of  the 
cross-section  to  which  it  belongs  will  obviously  be  ON,  i.e., 
at  right  angles  to  OQ. 

It  is  a  well-known  property  of  the  ellipse  that  the 
square  of  the  perpendicular  p  drawn  from  the  center  to 
the  tangent  to  the  curve,  if  the  inclination  of  that  per- 
pendicular to  the  semi-axis  is  a,  is ; 

£2=ri2cos2a+r22sin2a.        .     .     .     (10) 

This  value  of  p2  is  precisely  the  same  as  r'2  in  eq.  (7) 
and  it  shows  that  the  radius  of  gyration  OR  =  OD  about 
any  semi-diameter  OQ  considered  as  an  inertia  axis  is  equal 
to  the  normal  distance  between  that  semi-diameter  and  the 
parallel  tangent  RL' '.  This  simple  result  finds  an  import- 
ant application  in  the  problem  of  the  flexure  of  a  beam  of 
unsymmetrical  cross-section. 

This  same  normal  distance  between  a  semi-diameter 
of  the  ellipse  and  the  parallel  tangent  RL'  is  also  equal  to 

^,  the  semi-major  axis  of  the  ellipse  being  represented 

by  r\  and  the  semi-minor  axis  by  7-2,   while  r'  represents 
the  semi-diameter. 

The  preceding  equations  indicate  the  principal  proper- 
ties of  every  form  of  cross-section  which  may  affect  the  value 
of  the  moment  of  inertia  about  any  axis  whatever  passing 
through  its  centre  of  gravity. 


Art.  82.]          GORDON'S  FORMULA  FOR  LONG  COLUMMS. 


481 


a 


Art.  82. — Gordon's  Formula  for  Long  Columns. 

Since  flexure  takes  place  in  a  long  column  subjected  to 
a  thrust  in  the  direction  of  its  length,  the  greatest  intensity 
of    stress    in    a    normal    section    of    the    column  may  be 
considered  as  composed  of  two  parts, 
one  a   uniform   compression    over    the 
whole  section  the  total  of  which  is  equal 
to    the    load    on    the  column,  and  the 
other  the  usual  uniformly  varying  stress 
due  to  flexure  the  total  of  which  is  zero 
and  the  intensity  of  which  is  also  zero      *  j  p> 

along  the  neutral  axis  of  the  section. 
Fig.  i,  which  is  supposed  to  represent 
a  longitudinal  axial  section  of  a  column, 
shows  completely  this  composite  stress. 
The  line  fg  is  the  trace  of  the  normal 
section  and  gd=cf  =  p'  is  the  uniform 
intensity  of  compression  due  to  the  com- 
pressive load  P.  The  bending  moment  FIG.  i. 
is  represented  by  the  stresses  of  flexure 
varying  uniformly  in  intensity  from  p"  on  the  right-hand  side 
of  the  section  to  ef  on  the  left  side,  0  being  at  the  neutral 
axis.  The  compressive  stresses  are  indicated  by  —  and 
the  tensile  stresses  by  +.  The  resultant  of  these  two  com- 
posite stresses  is  a  uniformly  varying  stress  with  the  great- 
est compressive  intensity  p' +p"  on  one  side  of  the  section 
and  the  small  compressive  intensity  ec  on  the  left  side. 
The  bending  tension  neutralizes  exactly  the  same  amount 
of  uniform  compression,  making  the  resultant  intensity 
uniformly  varying.  There  is  no  resultant  tensile  stress  in 
the  section,  but  it  is  obvious  that  there  would  be  if  the 
bending  moment  were  sufficiently  large.  In  that  case  fe 
would  be  larger  than  jc.  This  condition,  however,  seldom 


482  LONG  COLUMNS.  [Ch.  X. 

occurs  in  actual  structural  columns  and  never  unless  they 
are  slender  and  too  heavily  loaded. 

The  condition  of  stress  as  described  above  is  that  ordi- 
narily assumed  for  columns,  but  the  actual  condition  of 
stress  is  frequently,  if  not  almost  invariably,  much  more 
complicated.  The  details  and  the  different  main  parts  of 
columns  do  not  act  with  perfect  concurrence  nor  are  the 
processes  of  manufacture  even  in  the  most  careful  shops 
such  as  to  leave  the  finished  members  without  internal 
stresses,  nor  are  they  perfectly  straight.  In  fact  the  best 
of  columns  may  be  a  little  convex  in  one  direction  at  one 
part  of  their  length  and  concave  in  the  same  direction  at 
another  part.  It  is  imperative,  however,  to  have  some 
reasonably  simple  rational  analysis  on  which  formulae  may 
be  based  leaving  the  erratic  stress  conditions  which  are  too 
obscure  and  uncertain  to  be  reached  by  analysis  to  be 
covered  by  empirical  coefficients  determined  by  tests  of 
actual  full-size  columns  and  the  stress  assumptions  illus- 
trated in  Fig.  i  fulfill  this  requisite  at  least  reasonably. 

In  order  to  determine  the  two  parts  of  the  resultant 
stresses  shown  in  Fig.  i,  let  5  represent  the  area  of  the 
normal  section;  I,  its  moment  of  inertia  about  a  neutral 
axis  normal  to  the  plane  in  which  flexure  takes  place;  r, 
its  radius  of  gyration  in  reference  to  the  same  axis;  P,  the 
magnitude  of  the  imposed  thrust;  /,  the  greatest  intensity 
of  stress  allowable  in  the  column,  and  J,  the  deflection 
corresponding  to  /.  Let  p'  be  that  part  of  /  caused  by  the 
direct  effect  of  P,  and  p"  that  part  due  to  flexure  alone. 
Then,  if  h  is  the  greatest  normal  distance  of  any  element 
of  the  column  from  the  axis  about  which  the  moment  of 
inertia  is  taken,  by  the  "  common  theory  of  flexure," 


Art.  82.]  GORDON'S  FORMPLA  FOR  LONG  COLUMNS.  483 

If  the  column  ends  are  round,  c9  =  i  ;  but  if  the  ends 
are  fixed,  the  value  of  c'  will  depend  upon  the  degree  of 
fixedness. 

Also 

P 

^-_;         .,     //  +  /^/a» 


Hence 

JS 


1  "f" 


(3) 


Eq.  (3)  may  be  considered  one  form  of  Gordon's  formula. 

In  order  to  make  eq.  (3)  workable  in  actual  computa- 
tions, it  is  necessary  to  express  the  deflection  A  in  terms 
of  known  dimensions  of  the  column.  By  referring  back 
to  eq.  (6a)  in  Art.  27  the  desired  expression  for  the  deflec- 
tion may  be  found  and  by  its  aid,  introducing  the  notation 
of  this  article,  eq.  (4)  may  be  at  once  written; 

a'p"J2        P  ,-, 

- 


It  is  seen,  therefore,  that  the  quantity  ax  depends  upon 
both  p"  and  E,  but  it  is  ordinarily  considered  constant. 
Since  I  =  Sr\  eqs.  (i)  and  (7)  give 


(5) 


Eq.  (8)  shows  that  a^^a. 
Hence 


484  LONG  COLUMNS.  [Ch.  X 

The  integration  by  which  eq.  (4)  is  obtained,  being 
taken  between  limits,  causes  everything  to  disappear 
which  depends  upon  the  condition  of  the  ends 
of  the  column.  Consequently  eq.  (6)  applies 
to  all  columns,  whether  the  ends  are  rounded 
or  fixed.  Let  the  latter  condition  be  assumed, 
and  let  it  be  represented  in  the  adjoining  figure. 
Since  the  column  must  be  bent  symmetrically, 
there  must  be  at  least  two  points  of  contraflexure. 
Two  such  points  only  may  be  supposed,  since 
such  a  supposition  makes  the  distance  between 
any  two  adjacent  points  the  greatest  possible 
and  induces  the  most  unfavorable  condition 
of  bending  for  the  column.  FIG.  2. 

If  B  and  C  are  the  points  of  contraflexure  supposed, 
then  BC  will  be  equal  to  a  half  of  AD,  for  each  half  of  BC 
must  be  in  the  same  condition,  so  far  as  flexure  is  concerned, 
as  either  A  B  or  CD.  Also  the  bending  moment  at  the 
section  midway  between  B  and  C  must  be  equal  to  that 
at  A  or  D.  Consequently  the  hinge-  or  round-end  column 
BC  must  possess  the  same  resistance  as  the  fixed-  or  flat- 
end  column  AD.  In  eq.  (6),  therefore,  let  /  =  2BC  =  2/1, 


W 


Eq.  (7)  is,  consequently,  the  formula  for  free-  or  round- 
end  columns  with  length  /i. 

The  flat-  or  fixed-end  column  AD  is  also  of  the  same 
resistance  as  the  column  AC,  with  one  end  flat  and  one 
end  round.  Hence  in  eq.  (6)  let  there  be  put  l 
and  there  will  result,  nearly, 


Art.  82.]  GORDON'S  FORMULA  FOR  LONG  COLUMNS.  485 


1+1.80^5 

Eq.  (8)  is,  then,  the  formula  for  a  column  with  one  end 
flat  and  the  other  round.  A  slight  element  of  approxima- 
tion will  ordinarily  enter  eq.  (8)  on  account  of  the  fact  that 
C  is  not  found  in  the  tangent  at  A  just  as  eqs.  (6)  and  (7) 
are  based  on  the  supposition  that  A  and  D  lie  exactly  in  the 
line  of  action  of  the  imposed  load. 

Eqs.  (7)  and  (8)  have  been  and  are  now  generally 
accepted  as  representing  the  resistances  of  columns  with  the 
end  conditions  to  which  they  are  intended  to  apply.  As  a 
matter  of  fact,  however,  tests  of  full-size  members  have 
demonstrated  that  those  conditions  are  not  realized  in  the 
actual  use  of  columns.  They  have  further  shown  that 
essentially  but  one  condition  of  column  ends  need  be 
recognized,  and  that  is  the  actual  pin-end  condition,  as 
realized  in  pin-connected  structures.  In  that  condition 
the  end  of  the  column  is  not  free  to  turn  .  The  compression 
between  the  pin  and  the  metal  bearing  against  it  caused 
by  the  load  carried  by  the  column  creates  a  considerable 
surface  of  contact  over  a  substantial  portion  of  which  the 
intensity  of  pressure  is  high.  This  produces  a  condition  of 
great  frictional  resistance  to  any  motion  between  the  pin  and 
the  end  of  the  column,  but  not  sufficient  probably  to  induce 
a  fixed-end  condition.  It  has  been  found  by  test  that  flat- 
end  columns,  as  a  rule,  give  less  ultimate  resisting  capacity 
than  pin-end  columns  of  the  same  length  and  same  radius 
of  gyration  of  cross-section.  This  is  doubtless  due  to  the 
practical  impossibility  to  secure  a  central  application  of 
loading  when  flat  ends  are  employed,  the  resulting  eccen- 
tricity reducing  the  ultimate  carrying  capacity  of  the 
members.  While,  therefore,  the  classes  .of  columns  repre- 


486  LONG   COLUMNS.  [Ch.  X. 

sented  by  eqs.  (7)  and  (8)  are  still  recognized,  it  would 
be  more  rational  and  more  in  accordance  with  experience 
to  use  only  the  general  form  of  eq.  (6)  with  a  determined 
from  actual  pin-end  tests. 

Although  the  quantities/  and  a,  in  eqs.  (6),  (7),  and  (8) 
are  usually  considered  constant,  they  are  strictly  variable. 
Eq.  (4)  shows  that  a  is  a  function  of  p'f  +E.  It  is  by  no 
means  certain  that  p"  is  the  same  for  different  forms  of 
cross-section,  or  even  for  different  sections  of  the  same 
form.  While  the  modulus  of  elasticity  E  varies  slightly  it 
may  properly  be  taken  as  constant. 

Again,  the  greatest  intensity  of  stress,  /,  which  can 
exist  in  the  column  varies  not  only  with  different  grades  of 
material,  but  there  is  some  reason  to  believe  that  it  must 
also  be  considered  as  varying  with  the  length  of  the  column. 
The  law  governing  this  last  kind  of  variation,  for  many 
sections,  still  needs  empirical  determination.  It  is  clear, 
therefore,  that  both  /  and  a  must  be  considered  empirical 
variables. 

Since  /  and  a  are  to  be  considered  variable  quantities, 

p 
let  y  take  the  place  of  /  and  x  that  of  a;    also,  let  p=-~ 

w3 

represent  the  mean  intensity  of  stress.  Eq.  (6)  then  takes 
the  form 


in  which  c  =  l2+r2. 

In  eq.  (9)  there  are  two  unknown  quantities,  y  and  #, 
consequently  two  equations  are  required  for  their  deter- 
mination. If  two  columns  of  different  ultimate  resistances 
per  unit  of  section,  and  with  different  values  of  c,  are  broken 
in  a  testing  machine,  and  the  two  sets  of  data  thus  estab- 
lished separately  -inserted  in  eq.  (9),  two  equations  will 


Art.  82.]  GORDON'S  FORMULA  FOR  LONG  COLUMNS.  487 

result  which  will  be  sufficient  to  give  y  and  x.     Those  two 
equations  may  be  written  as  follows  : 

y=p'(i+c'x),    .     .....     (10) 

y=p"(i+c"x).       .....     (n) 

The  simple  elimination  of  y  gives 


Either  eq.  (10)  or  (n)  will  then  give  y. 

In  selecting  experimental  results  for  insertion  in  eq. 
(12),  care  should  be  taken  to  make  the  differences  p"  —  pf 
and  c'  —  c"  as  large  numerically  as  possible,  in  order  that 
the  errors  of  experiment  may  form  the  smallest  possible 
proportion  of  the  first. 

In  consequence  of  the  more  or  less  erratic  results  of 
tests  of  full-sized  columns,  if  two  or  more  pairs  of  values 
of  /  and  a  be  found  as  indicated  above,  they  will  not  agree 
with  each  other  and  some  of  them  may  differ  largely.  Con- 
sequently the  procedures  illustrated  by  eqs.  (10),  (n)  and 
(12)  are  not  sufficient  for  a  satisfactory  determination  of 
the  quantities  desired.  The  method  of  probabilities  has 
been  employed,  but  it  also  is  unsatisfactory  because  of  the 
small  number  of  tests  available  if  for  no  other  reason. 

The  usual  process  is  to  plot  the  results  of  tests  using  -  for 

a  horizontal  coordinate  and  the  mean  load  per  square  inch 
of  cross-section  of  column  for  the  vertical  ordinate.  The 
results  of  -a  series  of  tests  will  in  this  manner  be  represented 
graphically  by  a  more  or  less  extended  group  of  points 

depending  upon  the  range  of  -.     A  curved  or  straight  line, 


438 


LONG  COLUMNS. 


[Ch.  X. 


as  the  case  may  be,  is  then  drawn  through  such  a  plotted 
group  of  points  so  as  to  give  it  a  mean  position  among  them. 
The  quantities  /  and  a  are  then  so  determined  by  trial  as 
to  produce  a  curve  lying  as  close  as  possible  to  the  experi- 
mental curve  and  the  resulting  equation  will  then  be 
Gordon's  formula  for  that  particular  set  of  tests  or  type  of 
columns.  This  operation  is  fully  illustrated  and  will  be 
further  considered  in  the  next  article  in  connection  with 
a  series  of  tests  of  Phoenix  columns  and  columns  of  other 
shapes. 

The  accompanying  diagrams  represent  some  cross-sec- 
tions of  columns  which  have  been  much  used. 


Batten  Lattice 


Quetec  Bridge 


SQUARE 


PHOENIX 


AM.BR.CO. 


TOP  CHORD  LATTICED. 


Z-BAR. 


SQUARE-LATTICED. 


PLATE  AND  ANGLES 


There  are  a  large  number  of  other  sections  which  have 
also  been  employed  either  for  wrought  iron  or  steel  columns. 
For  large  columns  it  is  occasionally  necessary  to  build  up 
cross-sections  consisting  of  a  number  of  webs  and  angles, 
all  so  secured  to  each  other  as  to  act  as  a  unit.  The  Quebec 
Bridge  section  is  such  a  one. 

Occasionally  a  so-called  "  swelled  "  column,  i.e.  with  a 
considerably  enlarged  cross-section  at  and  in  the  vicinity 


Art.  82.]  GORDON'S  FORMULA  FOR  LONG  COLUMNS.  489 

of  the  centre  of  the  column  length,  the  outline  of  section 
gradually  but  not  uniformly  decreasing  from  the  centre 
towards  the  ends,  is  required.  A  formula  for  such  a 
column  similar  to  Gordon's  formula  may  be  written  for  a 
varying  moment  of  inertia,  but  it  is  too  complicated  to  be 
of  practical  use.  In  the  case  of  such  columns  the  judgment 
of  the  engineer  must  be  used  in  applying  a  column  formula, 
but  it  will  generally  be  sufficient  to  take  the  radius  of 
gyration  at  the  middle  section  of  such  a  member  in  com- 
puting the  ratio.  — . 

The  preceding  formulae  and  the  considerations  on  which 
they  are  based  imply  without  qualification  that  all  parts 
of  a  column  must  be  so  rigidly  bound  together  that  each 
such  member  will  act  as  a  perfect  unit  under  loading  and 
they  include  the  condition  that  the  cross-section  of  the 
column  is  maintained  in  its  proper  shape  and  proportions 
without  material  distortion  up  to  actual  failure  of  the  tested 
columns.  It  is  imperative,  therefore,  in  the  design  of  these 
members  that  the  details,  including  rivets,  lattice  bars, 
batten  plates  and  other  spacing  details,  shall  be.  sufficient 
in  number  and  dimensions  to  maintain  the  column  as  a 
unit  up  to  its  full  carrying  capacity.  A  failure  to  meet 
these  conditions  may  greatly  and  perhaps  fatally  reduce 
the  carrying  capacity  of  the  column  and  result  in  disaster, 
as  in  the  case  of  the  first  Quebec  Bridge,  caused  by  the 
weak  latticing  of  a  compression  member.  If  a  column  more 
or  less  weak  in  its  spacing  or  other  details  is  tested  to  its 
ultimate  resistance,  it  will  yield  in  some  of  its  weak  details 
instead  of  failing  as  a  whole,  i.e.,  as  a  unit. 

The  general  principles  which  govern  the  resistance  of 
built  columns  may,  then,  be  summed  up  as  follows. 

The  material  should  be  disposed  as  far  as  possible  from 
the  neutral  axis  of  the  cross-section,  thereby  increasing  r; 


4QO  LONG  COLUMNS.  [Ch.  X. 

There  should  be  no  initial  internal  stress; 

The  individual  parts  of  the  column  should  be  mutually 
supporting; 

The  individual  parts  of  the  column  should  be  so  firmly 
secured  to  each  other  that  no  relative  motion  can  take  place,  in 
order  that  the  column  may  fail  as  a  whole,  thus  maintaining 
the  original  value  of  r. 

These  considerations,  it  is  to  be  borne  in  mind,  affect  the 
resistance  of  the  column  only;  it  may  be  advisable  to 
sacrifice  some  elements  of  resistance,  in  order  to  attain 
accessibility  to  the  interior  of  the  compression  member,  for 
the  purpose  of  painting.  This  point  may  be  a  very  im- 
portant one,  and  should  never  be  neglected  in  designing 
compression  members. 

Art.  83. — Tests  of  Wrought  Iron  Phoenix  Columns,  Steel  Angles 
and  Other  Steel  Columns. 

During  the  period  of  use  of  wrought  iron  as  a  struc- 
tural material  many  full-size  wrought-iron  columns  were 
tested  to  failure  giving  data  on  which  to  base  long  column 
formulae,  but  as  yet  few  steel  columns  of  full  size  have 
been  tested  to  failure  and  the  data  on  which  to  base  proper 
long  column  formulas,  either  for  ordinary  structural  carbon 
steel  or  for  nickel  steel,  are  correspondingly  meagre.  At 
this  time  (1915)  full-size  steel  columns  are  in  process  of 
testing  at  the  National  Bureau  of  Standards,  Washington, 
D.  C.,  and  when  they  are  completed,  the  desired  data  will 
be  much  increased. 

In  view  of  this  condition  of  experimental  work  on  steel 
columns  it  seems  best  to  give  the  results  of  tests  of  an 
extended  series  of  wrought  iron  Phoenix  columns  made  with 
much  care  at  the  U.  S.  Arsenal  at  Watertown,  Mass,  in 
order  to  illustrate  fully  the  method  of  graphical  treatment 


Art.  83. 


TESTS  OF  VARIOUS  STEEL  COLUMNS. 


491' 


of  such  results  in  the  process  of  seeking  proper  column 
formulae.  The  complete  account  of  this  series  of  tests  is 
given  in  the  Transactions  of  the  American  Society  of  Civil 
Engineers  for  1882  and  the  numerical  data  relating  both 
to  the  dimensions  of  the  columns  and  to  the  results  of  the 
tests  are  given  in  Table  I.  It  will  be  noticed  that  the  ratio 

TABLE  I. 


No. 

Length. 

Area. 

r*. 

/-s-r. 

/2-s-r2. 

E.L. 

Exp. 

Pi- 

pr. 

P". 

Feet. 

Sq.  In. 

Ins. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

i 

28 

12.062 

8.94 

112 

12,544 



35,i5o 

32,550 

34,488 



2 

28 

12.181 

8-94 

112 

12,544 



34,i5o 

32,550 

34,488 

„ 

3 

25 

12.233 

8-94 

Q   „  . 

IOO 

10,000 

27,960 

35,270 

34,000 

35,040 



4 

2S 

22 

I  2  .  IOO 

1^.371 

a  .  94 
8.94 

8_  . 

88 

QO 

7,744 



35,040 
35,570 

34iOOO 

35,420 

35,040 
35,592 

zz 

7 

22 
19 

12.311 

12  .023 

.  94 

8.94 

OO 

76 

7,744 
5,776 



34,360 
35,365 

35,420 
36,800 

35,592 
36,144 



8 

19 

12.087 

8.94 

76 

v  5,776 

29,290 

36,900 

36,800 

36,144 



9 

16 

12  .000 

8.94 

64 

4,096 

36,580 

38,130 

36,696 



10 

16 

I  2  .  OOO 

8.94 

64 

4,096 



36,580 

38,130 

36,696 



ii 

13 

12.  iSS 

8.94 

52 

2,704 

28,890 

36,857 

39,400 

37,248 



12 

13 

I  2  .  069 

8.94 

52 

2,704 



37,200 

39,400 

37,248 



13 

10 

12.248 

8-94 

40 

i,  600 

26,940 

36,480 

40,700 

37,8oo 

. 

14 

10 

12.339 

8.94 

40 

i  ,600 

28,360 

36,397 

40,700 

37,8oo 



15 

7 

12.  265 

8.94 

28 

784 

29,350 

38,157 

42,200 

38,352 

40,360 

16 

7 

II  .  962 

8.94 

28 

784 

29,590 

43,3oo 

42,200 

38,352 

40,360 

i? 

4 

12.081 

8.94 

16 

256 

49,500 

44,770 

46,300 

18 

4 

12.119 

8.94 

16 

256 

28,050 

51,240 

44,770 



46,300 

19 

3  ins. 

11.903 

8.94 

2-7 

7.29 



57,130 

69,600 



57,140 

20 

8  ins. 

II  .903 

8.94 

2.7 

7.29 



57,300 

69,600 



57,140 

21 

25'  2.65" 

18.  300 

19-37 

68.8 

4,733 



36,010 

37,600 

36,666 

22 

8'  9" 

18.  300 

19-37 

24 

576 

29,510 

42,180 

42,840 

42,160 

of  length  over  radius  of  gyration  ranges  from  less  than  3  up 
to  1 1 2  which  more  than  includes  the  values  of  that  ratio 
in  practically  all  steel  structural  work.  The  ends  of  the 
columns  were  flat,  a  condition  which  usually  introduces 
some  erratic  results,  but  apparently  the  care  with  which 
the  columns  were  tested  eliminated  this  defect.  The 
Phoenix  column  is  a  particularly  advantageous  section  for 
testing  as  its  different  parts  are  effectively  self-supporting 
and  furthermore  it  has  a  section  whose  radius  of  gyration 
is  the  same  in  all  directions  as  the  latter  has  two  or  more 
axes  of  symmetry  not  at  right  angles  to  each  other. 

The  numerical  quantities  in  Table  I  are  self-explanatory, 


492 


LONG   COLUMNS. 


[Ch.  X. 


particularly  in  connection  with  eq.    (6)   of  the  preceding 
article. 

The  five  columns  in  the  right-hand  half  of  the  Table 
are  pounds  per  square  inch  for  the  different  purposes  shown 
by  the  headings  of  the  columns,  i.e.  E.  L.  represents  the 
compressive  stress  in  the  column  at  the  elastic  limit, 
while  the  column  headed  Exp.  indicates  the  compressive 
load  per  square  inch  of  section  at  which  it  failed  in  the 
testing  machine.  The  headings,  pi,  p'  and  p"  are  computed 


72000 


12000 


60 


FIG.  i. 


values  from  eqs.  (i),  (2),  and  (3)  to  be  explained  immedi- 
ately. 

The  numerical  values  in  the  column  headed  Exp.  are 
accurately  plotted  in  the  diagram,  Fig.  i,  by  laying  off  the 

ratios  -  from  0  to  the  left  as  horizontal  ordinates  and  erect- 
r 

ing  at  their  extremities  the  corresponding  ultimate  resist- 
ances given  in  that  column  as  vertical  ordinates  with  the 
scale  as  shown  in  Fig.  i.  It  should  be  observed  that 
in  the  majority  of  cases  in  Table  I,  there  are  two  experi- 
mental results  for  each  value  of  and  each  vertical  or  din  ate 

r 

in  Fig.  i  represents  the  mean  of  these  two  results. 


Art.  83.]  TESTS  OF   VARIOUS  STEEL  COLUMNS.  493 

The  full-curved  line  marked  "  Watertown  Exp.  Curve  " 
is  then  drawn  so  as  to  represent  as  accurately  as  possible 
the  actual  experimental  results  which,  as  shown  in  the 
figure,  include  a  few  tests  other  than  those  made  at  Water- 
town.  This  experimental  curve  rises  rapidly  for  small 

values  of  -,  i.e.,  for  what  are  actually  short  blocks.     At 
r 

the  left  end  of  the  curve  where  -  equals  140,  the  slope  of  the 

curve  is  but  little  more  than  for  intermediate  values  of.  that 
ratio. 

After  a  number  of  trials  it  was  found  that  the  value  of 
piy  as  given  in  eq.  (i),  agrees  quite  closely  with  the  experi- 

mental curve  for  all  values  between  -  =  28  and  -  =  112,  and 

r  r 

the  results  computed  from  it  are  shown  in  the  column  headed 
pi  of  the  Table 


+— 

(i) 


50000  r2 

Eq.  (i)  is  Gordon's  formula  for  this  particular  set  of 
Phoenix  columns  except  that  the  value  of  /  (the  numerator 
of  the  second  member)  is  seen  to  vary  slightly  with  the 

A* 

ratio  -.     In    actual    engineering    practice,    however,    the 

numerator  shown  in  eq.  (i)  was  displaced  by  the  numerical 
value  42,000,  as  a  constant  numerator  of  the  second  member 
makes  a  simpler  application  of  the  formula  and  it  was 
sufficiently  accurate  for  all  practical  purposes. 

Inasmuch  as  all  long  columns  used  in  structural  work  are 

found  within  the  limits  of  -=30  and  -  =  120  (usually  for 


494  LONG  COLUMNS.  [Ch.  X. 

bridge  truss  members,  100)  Gordon's  formula  is  never  used 
outside  of  practically  these  limits. 

It  may  be  observed  that  the  experimental  curve  is 
nearly  a  'straight  line  from  a  point  just  above  b  to  the 
extreme  left  of  the  diagram.  For  that  portion  of  the 
curve,  therefore,  the  following  formula  applies  very  closely: 

£'=39,640 -46-.* (2) 

The  results  of  this  formula  are  given  in  the  column 
headed  "  p'."  The  table,  in  connection  with  the  diagram, 
shows  that  this  formula  may  be  used  with  accuracy  for 
values  of  l  +  r  lying  between  30  and  140,  and  further  ex- 
periments may  possibly  show  that  it  is  applicable  above 
the  latter  limit. 

For  values  of  l-r-r  less  than  30,  the  following  formula 
will  be  found  to  give  results  approximating  very  closely  to 
the  experimental  curve: 

p"  =  64,700— 4, ( 


The  results  of  the  application  of  this  formula  are  given 
in  the  column  headed  "  pn '." 

It  will  be  observed  in  Table  I  that  the  ultimate  resist- 
ance per  square  inch  of  the  Phoenix  columns  tested  for 

*  This  equation  known  as  the  straight-line  formula  for  long  columns  was 
first  proposed  in  a  paper  by  the  author  before  the  Annual  Convention  of  the 
American  Society  of  Civil  Engineers  in  1881.  It  was  established  at  that 
time  concurrently,  but  independently,  by  the  author  and  Prof.  Mansfield 
Merriman.  The  formula  is  sometimes  called  the  Johnson  Straight  Line 
Formula,  but  Mr.  Johnson's  paper,  in  which  he  discussed  the  straight-line 
formula,  was  not  given  to  the  American  Society  of  Civil  Engineers  until 
1885,  four  years  after  the  papers  by  the  author  and  Professor  Merriman  had 
been  published. 


Art.  83.]  TESTS  OF  VARIOUS  STEEL  COLUMNS.  495 

ratios  of  -  between  about  40  and  112  ranges  from  about 

34,000  to  about  38,000  pounds,  which  is  somewhat  above 
the  yield  point  of  the  material  but  far  below  the  ultimate 
compressive  resistance  per  square  inch  as  found  for  short 
blocks. 

In  built-up  sections  of  columns  in  which  the  component 
parts  are  less  well  supported  than  in  the  Phoenix  section, 
the  ultimate  column  resistance  per  square  inch  will  be  but 
little  if  any  above  the  yield  point  of  the  material  and  with 

high  values  of  -  the  ultimate  resistance  may  not  rise  above 

the  elastic  limit.  This  is  a  most  important  feature  of  long 
column  resistance  and  it  shows  the  effect  of  bending  or 

flexure  which  increases  as  -  becomes  greater. 

r 

Many  tests  of  full-size  pin-end  wrought-iron  columns 
have  shown  that,  when  well  designed  with  lattice  bars  and 
other  spacing  details  of  sufficient  capacity,  the  ultimate 
resistance  of  such  columns  may  be  represented  by  eqs.  (4) 
and  (40); 


30,000  r2 
Or; 


£=42,500-140-  ......     (40) 


Although  either  equation  is  for  columns  with  pin  ends, 
it  may  be  used  generally  for  such  end  conditions  as  are 
usually  found  in  structures  like  bridges  or  buildings.  The 
flat  end  condition  has  already  been  indicated  as  giving  in 
general  somewhat  erratic  results,  but  with  no  advantage 


496  LONG  COLUMNS.  [Ch.  X. 

over  pin  ends  for  ordinary  circumstances  or  for  such  ratios 
of  -  as  are  commonly  employed. 

For  working  stresses  in  wrought-iron  columns  eqs.  (5) 
or  (50)  may  be  used.  They  are  derived  from  eqs.  (4)  or 
(40)  by  dividing  the  second  members  of  those  equations 
by  a  so-called  "  safety  factor  "  of  about  3.5;  . 

-       "'0°%,  (5) 


i 
H 


30,000  r2 


Or; 

£  =  I2,OOO—4O- (50) 


Steel  Columns. 

The  paucity  of  tests  of  suitably-designed  full-size  steel 
columns,  either  with  pin  ends  or  other  end  conditions,  has 
already  been  observed.  Some  scattered  tests  of  such  mem- 
bers have  fortunately  been  made  while  others  have  been 
made  upon  members  so  designed  as  to  bring  out  in  ex- 
aggerated form  certain  features  of  actions  of  stresses  in 
various  parts  of  the  columns  without,  however,  reaching 
data  available  for  the  best  designs  for  general  engineering 
practice. 

Among  the  most  valuable  of  these  data  are  some  results 
of  old  tests  by  the  late  Mr.  James  Christie  and  described 
in  the  Transactions  of  the  American  Society  of  Civil  Engi- 
neers for  1884.  Mr.  Christie  tested  mild  and  high  steel 

angle  struts  with  ratios  of  -  running  from  20  up  to  300. 

The  mild  steel  contained  from  .11  to  .15  per  cent,  carbon, 
while  the  high  steel  contained  .36  per  cent.  The  ultimate 
tensile  resistance  of  the  mild  steel  ran  from  60,000  to  66,000 


Art.  83.] 


TESTS  OF   VARIOUS  STEEL  COLUMNS. 


497 


pounds  per  square  inch  with  24  to  26  per  cent,  stretch  in 
8  inches.  The  high  steel  had  an  ultimate  tensile  resistance 
of  about  100,000  pounds  per  square  inch  and  a  stretch 
of  about  1 6  per  cent,  in  8  inches. 

Table  II  gives  the  results  of  these  steel  angle  tests  and 


TABLE    II. 
FLAT-END  STEEL  ANGLE  STRUTS. 


Ultimate  Resistance,  Pounds 

Ultimate  Resistance,  Pounds 

1 

per  Square  Inch. 

I 

per  Square  Inch. 

r 

Mild  Steel. 

High  Steel. 

r 

Mild  Steel. 

High  Steel. 

20 

72,000 

100,000 

170 

21,000 

26,000 

30 

51,000 

74,000 

1  80 

19,500 

23,800 

40 

46,000 

65,000 

190 

l8,000 

2I,8oo 

50 

43,000 

61,000 

2OO 

16,500 

2O,OOO 

60 

41,000 

58,000 

210 

15,200 

18,400 

70 

39,000 

56,000 

220 

14,000 

16,900 

80 

38,000 

54,000 

230 

13,000 

15,400 

90 

36,500 

51,000 

240 

I2,OOO  " 

14,000 

100 

35,ooo 

47,000 

250 

11,100 

12,800 

no 

33,500 

43,500 

260 

10,300 

1  1,  800 

1  20 

3!,500 

40,000 

270 

9,600 

II,OOO 

130 

29,000 

36,500 

280 

9,000 

10,200 

140 

27,000 

33,500 

290 

8,400 

9,500 

150 

25,000 

30,800 

300 

7,900 

9,000 

•    1  60 

23,000 

28,300 

Fig.  2  shows  the  curves  formed  by  plotting  in  the  usual 
manner  the  ultimate  resistances  found  in  Table  II.  The 

ratios  -  are  laid  off  as  horizontal  ordinates  and  the  corre- 
r 

sponding  ultimate  resistances  as  vertical  ordinates.  These 
curves  are  highly  interesting  as  exhibiting  the  various 
stages  of  resistance  offered  by  columns  in  compression  as 

the  lengths  increase  from  small  values  of  -  up  to  large  values 
of  that  quantity.  The  ultimate  resistances  decrease  rapidly 


498 


LONG  COLUMNS. 


[Ch.  X. 


Art.  83. 


TESTS  OF  VARIOUS  STEEL  COLUMNS. 


499 


when  the  column  ratio  -  increases  from  20  to  40,  then  up 

to  a  value  of  at  least  140  the  curves  differ  but  little  from 
straight  lines.     Above  the  latter,   the  curvature  becomes 


decided  but  not  sharp  and  the  two  lines  converge  so  that 
when  -  becomes  equal  to  300  the  difference  between  the  two 
resistances  is  but  little  over  1000  pounds  per  square  inch.. 


500  LONG  COLUMNS.  [Ch.  X. 

This  convergence  is  one  element  of  confirmation  of  Euler's 
Formula  as  the  carrying  capacity  for  such  high  values  of 

-  depends  chiefly  upon  the  modulus  of  elasticity.     With 

still  higher  ratios  the  two  curves  would  probably  coincide 
as  both  grades  of  steel  have  the  same  modulus. 

The  difference  between  the  working  parts  of  the  two 
curves  shown  in  Fig.  2  is  reproduced  on  a  much  larger  scale 

for-  in  Fig.  3.     Between  -=30  and -=140,  the  two  full 

straight  lines  may  be  drawn  as  shown.  As  the  points 
represent  accurately  the  numerical  values  of  Table  II,  it 
is  seen  that  the  straight  lines  represent  the  ultimate  resist- 
ances of  the  angle  struts  with  sufficient  closeness  for  all 

practical  purposes  between  -  =35  and  -  =  140. 

The  straight  line  for  the  mild-steel  angles  is  represented 
by  eq.  (6) ; 

p  =  53,000  - 186  -.       ...'..     (6) 

Similarly  the  straight  line  for  the  high-steel  angles  is 
represented  by  eq.  (7) ; 

£  =  79,000-325- (7) 

The  curved  broken  lines  represent  approximately  the 
unit  ultimate  resistances  for  -  less  than  about  40.     If  the 

second  members  of  eqs.  (6)  and  (7)  be  divided  by  a  so-called 
"  safety  factor  "  of  about  3,  eqs.  (8)  and  (9)  will  represent 
working  stresses; 

For  high  steel       £  =  25,000  —  100- (8) 

For  mild  steel       £  =  17,000  —  53-.  .     .     .     .     .     .     (9) 


Art.  83.]  TESTS  OF  VARIOUS  STEEL  COLUMNS.  501 

A  number  of  "  model  "  carbon  steel  columns  of  large 
dimensions  have  been  tested  within  two  or  three  years  in 
the  large  testing  machine  of  the  Phoenix  Bridge  Company 
at  Phoenixville,  Pa.,  together  with  two  such  nickel  steel 
columns,  under  the  supervision  of  Mr.  James  E.  Howard, 
all  but  three  of  those  tests  having  been  made  for  the  pur- 
pose of  affording  data  for  the  design  of  the  new  Quebec 
Bridge  across  the  St.  Lawrence  River.  The  results  of  these 
tests,  as  given  in  the  Transactions  of  the  American  Society 
of  Civil  Engineers  for  1911  and  in  the  Engineering  Record 
for  1914  are  shown  on  Fig.  3.  The  average  of  three  tests 
of  built  up  carbon  steel  columns,  30  inches  by  20  inches 


a 

42.75  Sq.  Ins. 


b 

66.65  Sq.  Ins.J 


n 
34.63  Sq.  Ins. 


FIG   4.  FIG.  5.  FIG.  6. 

in  outline,  as  indicated  by  Fig.  4,  are  shown  at  d,  the  value 
of  --  being  47  and  the  average  ultimate  resistance  of  the 

three  tests  (varying  but  little  from  each  other)  being  30,000 
pounds  per  square  inch. 

The  results  shown  at  e,  f  and  g  are  also  for  carbon 
steel  columns  with  built-up  sections  shown  in  the  diagram 
on  page  488,  the  cro^s-sectional  area  being  70.65  square 
inches.  The  length  of  these  columns  was  18  feet  9  inches 

and  the  ratio  -  was  38. 

Again  a  and  b  represent  results  for  carbon-steel  columns 
having  -  equal  to  78  and  58  and  with  cross-sectional  areas 
42.75  square  inches  distributed  as  shown  in  Fig.  5.  , 


502  LONG  COLUMNS.  [Ch.  X. 

Finally,  the  point  n  represents  the  result  for  two  nickel 
steel  columns  having  an  area  of  cross-section  of  34.63  square 

inches  and  -  =  52,  the  section  being  shown  in  Fig.  6/ 

The  number  of  tests  of  the  carbon-steel  columns  is  not 
sufficient  to  form  a  proper  basis  for  a  straight  line  long 
column  formula,  but  the  broken  line  drawn  through  a  and 
c  and  below  e  may,  as  a  tentative  matter,  be  represented 
by  eq.  (10); 

£=44,000  —  150- (10) 

All  these  built-up  carbon-steel  columns  were  of  mild 
steel,  but  their  ultimate  resistances  are  distinctly  lower  than 
the  results  for  Mr.  Christie's  mild-steel  angles.  Full-size 
tests,  however,  have  shown  that  the  built-up  column,  unless 
designed  with  great  care  so  as  to  act  solidly  as  a  unit,  will 
not  offer  ultimate  resistances  as  high  as  might  be  expected 
from  the  quality  of  the  steel  of  which  they  are  composed. 

On  the  same  basis  used  for  eqs.  (7)  and  (9),  the  tentative 
working  stress  for  built-up  mild  carbon-steel  columns  would 
be; 

£  =  14,000-50    .......          (ll) 

The  average  for  the  two  nickel-steel  columns,  shown  at 
n,  Fig.  3  is  about  50,000  pounds  per  square  inch  and  more 
than  one-third  greater  than  the  corresponding  result  shown 
for  the  mild  carbon  steel  at  c. 

In  all  these  column  tests  the  elastic  limit  or  the  yield 
point  of  the  member  as  a  whole  appears  to  be  the  controlling 
feature,  i.e.,  the  ultimate  resistance  is  not  above  the  yield 

point  of  the  column  and  if  the  ratio  -  is  comparatively  large 

it  will  not  be  above  the  limit  of  elasticity  of  the  column  as 
a  whole.     It  must  be  remembered  also  that  both  the  elastic 


Art.  83.]  TESTS  OF  VARIOUS  STEEL   COLUMNS.  503 

limit  and  the  yield  point  of  built-up  columns  will  be 
materially  lower  than  the  corresponding  points  of  a  single 
piece  of  the  same  metal. 

These  tests  appear  to  indicate  that  the  ultimate  resistances 
of  nickel-steel  columns  exceed  those  of  mild  carbon-steel 
columns  in  about  the  same  proportion  that  the  elastic  limit 
of  nickel  steel  exceeds  the  elastic  limit  of  the  carbon-steel. 

Observations  in  these  tests  of  full-size  columns  made 
at  Phoenix ville  by  Mr.  Howard  indicate  that  steel  columns 
may  be  considered  to  have  a  true  modulus  of  elasticity  of 
about  29,000,000  or  perhaps  29,500,000  for  intensities  of 
loading  not  greater  than  ordinarily  allowed  working  stresses, 
i.e.,  from  8,000  to  12,000  pounds  per  square  inch.  While 
there  are  not  sufficient  data  to  determine  precisely  such 
physical  elements  of  steel  column  resistance,  there  seems 
to  be  a  relative  motion  of  the  component  parts  of  a  built-up 
member  under  test,  which  does  not  permit  the  existence 
of  a  true  modulus  of  elasticity  when  loadings  exceed  about 
12,000  to  15,000  pounds  per  square  inch.  Obviously  the 
more  nearly  a  column  acts  as  a  perfect  unit,  the  better 
defined  will  be  its  elastic  properties. 

Much  more  data  derived  from  experimental  work  with 
full-size  steel  columns  are  imperatively  necessary  in  order 
to  reach  definite  conclusions  regarding  actions  of  stresses 
in  the  various  parts  of  such  members  as  well  as  for  the 
development  of  such  important  details  as  latticing,  battens, 
and  other  riveted  details. 

Typical  Formula  Now  in  Use. 

As  a  result  of  the  present  conditions  of  experimental 
knowledge  of  built  columns,  as  well  as  of  those  that  are 
not  built  up,  there  is  a  great  variety  of  column  formulae 
used  by  engineers,  both  of  the  Gordon  and  straight-line 
type.  The  straight-line  formula,  however,  is  largely  dis- 


504  LONG  COLUMNS.  [Ch.  X. 

placing  the  Gordon  formula.  The  General  Specifications 
for  Steel  Railway  Bridges  recommended  by  the  American 
Railway  Engineering  Association  as  applied  to  the  design 
of  cross-sections  of  steel  columns  is; 

p  =  i6,  000  —  70-  ......     (12) 

The  New  York  Central  Lines  are  using  the  same  formula 
in  the  design  of  their  bridge  work,  as  are  engineering  or- 
ganizations of  other  railway  companies.  Under  the  use 
of  this  formula  a  greater  compressive  load  than  14,000 
pounds  per  square  inch  is  not  permitted. 

The  American  Bridge  Company  Specifications  for  Steel 
Structures  1913,  uses  the  following  formula  in  its  design  work  ; 

£  =  19,000  —  100  —  ......     (13) 

A  provision  for  impact  is  made  and  13,000  pounds  per 
sq.  in.  is  the  maximum  allowed  under  the  use  of  eq.  (13). 

A  form  of  Gordon's  formula  still  appearing  in  engineer- 
ing practice  is 

12,500 

p=  '  -----   (14) 


36,000  r2 

This  formula  is  really  an  old  wrought-iron  column 
formula  and  should  not  be  used  without  reducing  the  36,000 
in  the  denominator  to  30,000. 

The  New  York  Building  Law  gives  for  a  steel  column  ; 

£  =  15,200-58  -.      .     .     .     .    '.,    (15) 

The  formula  used  by  the  City  of  Philadelphia  for  its 
buildings  is  of  the  Gordon  type  as  follows: 

16,250 
P=-      —  -  J2  ......     (l6) 

iH  ---  .-« 
11,000  r2 


Art.  83.]  TESTS  OF  VARIOUS  STEEL  COLUMNS.  505 

Other  formulae  could  be  cited  but  enough  is  shown  to 
indicate  the  pronounced  lack  of  uniformity  in  this  practice. 

None  of  the  preceding  formulae  should  be  used  for  - 
less  than  30  nor  more  than  about  120. 

In  every  case  where  a  column  formula  is  used,  it  would 
be  much  more  convenient  to  employ  a  diagram  with  the 
curves  accurately  drawn  to  represent  the  desired  formulae. 
The  actual  results,  without  computations,  could  be  read 
directly  from  such  long  column  curves. 

Details  of  Columns. 

In  addition  to  the  data  already  given  in  another  portion 
of  this  article,  the  tests  cited  in  this  chapter  show  that 
the  unsupported  width  of  no  plate  in  a  compression  member 
should  exceed  30  to  3  5  times  its  thickness.  These  tests  have 
usually  been  made  with  plates  or  metal  J  to  ^  inch  in  thick- 
ness, and  it  is  altogether  probable  that  the  above  ratio 
of  width  over  thickness  would  be  increased  with  greater 
thicknesses. 

In  built  columns,  however,  the  transverse  distance  between 
centre  lines  of  rivets  securing  plates  to  angles  or  channels,  etc., 
should  not  exceed  35  times  the  plate  thickness.  If  this  width 
is  exceeded,  longitudinal  buckling  of  the  plate  takes  place, 
and  the  column  ceases  to  fail  as  a  whole,  but  yields  in  detail. 

The  same  tests  show  that  the  thickness  of  the  leg  of  an 
angle  to  which  latticing  is  riveted  should  not  be  less  than  %  of 
the  length  of  that  leg  or  side,  if  the  column  is  purely  and 
wholly  a  compression  member.  The  above  limit  may  be 
passed  somewhat  in  stiff  ties  and  compression  members 
designed  to  carry  transverse  loads. 

The  panel  points  of  latticing  should  not  be  separated  by  a 
greater  distance  than  60  times  the  thickness  of  the  angle  leg  to 
which  the  latticing  is  riveted,  if  the  column  is  wholly  a  com- 
pression member. 


LONG  COLUMNS. 


[Ch.  X. 


The  rivet  pitch  should  never  exceed  1 6  times  the  thickness 
of  the  outside  thinnest  metal  pierced  by  the  rivet,  and  if  the  plates 
are  very  thick  it  should  never  nearly  equal  that  value. 

Art.  84. — Complete  Design  of  Pin-end  Steel  Columns. 

In  actual  design  it  is  necessary  not  only  to  make  appli- 
cation  of  the  preceding  formulae  for  ultimate  resistance  of 
columns,  but  also  to  proportion  a  considerable  number  of 
details  as  matters  largely  of  judgment  and  experience.  If 
the  column,  like  the  section  shown  as  the  latticed  channel 
or  latticed  upper  chord  in  the  preceding  article,  has  two 
open  sides  as  in  the  former  or  one  open  side  as  in  the  latter 
latticed,  i.e.,  has  small  bars  of  iron  running  diagonally 
across  those  open  sides  in  order  to  hold  the  parts  of  the 
column  in  their  proper  relative  positions,  those  lattice 
bars  vary  in  size  with  the  size  of  column.  While  the  dimen- 
sions vary  somewhat  among  engineers,  the  following  table, 
which  has  been  largely  used,  illustrates  effectively  sizes 
that  may  properly  be  employed. 


"or     6     in 

7 
8 

9 
10 
ii 

12 
13 

15 

16 
18 
19-23 
24-29 

"       30 

ch  ro 

>r  bi 

< 
< 

« 

< 
« 
« 
< 

r             | 

a  =  --{ 

b 
fir  ii' 
"    if 

"     2 

ledc 

< 

< 

i 

< 

'             ,[ 

lilt  channels        if'XiV  " 

if    X-nr 

l^     X  T$ 

if  and  2     Xf 

'                 "                           if      "       2        X^ 

-.2        Xf 

.    2        Xf 

2i   Xf 

24r    Xf 

.2!  xf 

2*     Xf 

'                                                                  2i     Xf 

•                 '«                                              2!     V  1 

..^     X* 

a 
.1' 


Art.  84.]     COMPLETE  DESIGN   OF  PIN-END  STEEL  COLUMNS.        507 


These  bars  or  lattices  may  be  used  in  single  system,  in 
which  case  each  one  should  make  an  angle  of  about  60°  with 
the  centre  line  of  the  side  of  the  column  on  which  they  are 
placed.  If  they  are  used  in  double  system  each  pair  of 
bars  will  intersect  at  their  mid-points,  and  in  this  case  the 
bars  may  make  angles  of  45°  with  the  centre  line  of  the  side 
of  the  column  on  which  they  are  employed.  In  the  case 
of  double  latticing  the  intersecting  pairs  of  bars  are  riveted 
at  their  intersections.  Lattice  bars  are  held  at  their  ends 
by  one  rivet  or  by  two  rivets  according  to  the  size  of  the 
column,  as  shown  in  the  next  table. 

Figs,  i,  2,  and  3  illustrate  different  modes  of  riveting 
the  ends  of  lattice  bars.  The  size  and  number  of  rivets 


o  o  o  p_o_o_o  p 


oooooooo 


FIG.  i. 


FIG.  2.  FIG.  3. 

will  obviously  depend  upon  the  size  of  the  lattice  bars 
employed  and  to  some  extent  upon  the  manner  in  which 
their  ends  are  held. 

The  following  table  has  been  used  in  actual  structural 
practice  and  exhibits  good  practice  in  the  design  of  single 
latticing.  It  is  based  on  the  supposition  that  the  lattice 
bars  are  flats.  In  very  large  columns  or  in  some  exposed 


LONG  COLUMNS. 


[Ch.  X 


situations  it  is  necessary  to  use  steel  angles  for  latticing, 
the  ends  of  which  must  be  secured  by  rivets  proportionate 
in  number  and  diameter  to  the  size  of  angle. 


Size  of  Lattice. 

Rivets  :    Number 
and  Size. 

Number  of  Rivets 
at  Lattice  Point. 

Limiting  Length  of 
Lattice  Centre  to  Centre 
of  Inner  Rivets. 

if  XT*  and  f 

n 

I 

13  inches 

. 

I 

16 

.• 

I 

10 

I 

23 

I 

16 

.- 

i  or  2 

20 

I    "    2 

15 

2^  X  yV 

'\ 

I    "    2 

20 

2^  XT'? 

'« 

I    "    2f 

•17 

2$X£ 

I    "    2 

26 

. 

f 

1 

I    "    2 

24 

2^Xi 

4 

15 

3X1 

i  or  2 

18 

3X^ 

f 

I    "    2 

16 

3Xi 

. 

4 

9 

iX'Aj 

• 

i  or  2 

25 

3XT7* 

•   ^f 

I    "    2 

22 

-V 

. 

.f 

4 

15 

3Xi 

•1 

i  or  2 

32 

3Xi 

•If         - 

I    "    2 

29 

3X* 

2. 

.f 

4 

21 

3X4 

2. 

•i 

4 

II 

I  . 

•jf 

i  or  2 

28 

. 

4XrV 

2 

.f 

4 

22 

1 

2. 

1 

4 

15 

At  each  end  of  the  open  or  latticed  sides  of  the  column 
are  placed  batten  plates  which  limit  the  latticing.  The 
width  of  these  batten  plates  is  determined  evidently  by  the 
width  of  the  column,  but  the  lengths  vary  somewhat  under 
different  specifications.  A  good  and  convenient  rule  is 
to  make  the  length  of  a  batten  plate  at  least  equal  to 
its  width.  The  thickness  of  a  batten  plate  will  depend 
upon  the  size  of  column ;  it  is  seldom  made  less  than  |  in. 
and  usually  not  more  than  |-  in.  for  large  columns.  The 
size  of  rivet  will  also  depend  upon  the  size  of  columns, 
Rivets  less  than  f  in.  in  diameter  are  seldom  used  in  railroad 


Art.  84.]     COMPLETE  DESIGN  OF  PIN-END  STEEL  COLUMNS.         509 

work  and  rarely  more  than  i  in.,  the  prevailing  diameter 
being  J  in. 

One  of  the  most  important  details  of  a  column  is  the 
jaw  or  extension  of  one  side  at  the  end.  The  two  jaws 
contain  the  pin  holes  through  which  are  transferred  to  the 
pin  the  total  load  carried  by  the  column.  These  jaws  or 
extensions  are  formed  so  as  to  fit  in  between  the  parts  of 
intersecting  members,  usually  the  upper  or  lower  chords 
and  eye-bars.  It  is,  therefore,  imperative  to  make  them 
as  thin  as  the  bearing  upon  the  pins  and  the  carrying 
capacity  of  the  jaws  themselves  acting  as  short  columns 
will  permit.  Figs.  4,  5,  6,  and  7  exhibit  some  types  of 


FIG.  4. 

these  post  jaws  as  they  commonly  occur.  As  the  figures 
show,  they  are  formed  by  cutting  away  the  flanges  of  the 
angles  or  channels  forming  parts  of  the  posts  and  riveting 
on  the  pin  or  thickening  plates  required  to  strengthen 
the  detail.  The  jaws  form  short  columns  whose  lengths 
should  be  taken  from  the  centre  of  the  pin  hole  to  the  last 
centre  line  of  rivets  in  the  body  of  the  column  back  of  the 


LONG  COLUMNS. 


[Ch.  X. 


cut  in  the  angle  or  in  the  flange  of  the  channel.     This 
length  indicated  by  /  is  shown  in  each  of  the  figures. 
There  have  been  but  few  tests  made  to  determine  the 


o      c. 

U       v. 

'       U 

f~}        C 

£\ 

_ 

^F 

^  u 

O      ( 

\ 

r^& 

^  ^ 

\^l       ^ 

\J 

^ 

o     o   o     o   o 

O       f 

• 

1  

\ 

FIG.  5. 


O      Pi      Pi      O 

t 

-j       r 

j 

O      ( 

")       (1 

-} 

-- 

, 

-    f' 

j 

v 

1   $ 

§5 

\    \0>, 

<j 

-1       ' 

| 

©  ^ 

1 

O     O     O     O 

-f 

=t  ^ 

%  —  ( 

=>  —  f 

1       > 

1 

"T 

\ 

r._ 

n 

E-r 

'-ZT- 

z.z 

~~f"~ 

-^Y^ 

~- 

1 

FIG.  6. 


C1TNTFR  LIME  CF  POST 


FIG.  7. 

resisting  capacities  of  this  particular  detail,  but  those  which 
have  been  made  form  the  basis  of  the  following  formula 
for  'medium  steel  columns.  Obviously  there  will  usually 


Art.  84.]     COMPLETE  DESIGN  OF  PIN-END  STEEL  COLUMNS.         511 

be  at  least  two  jaws  at  the  end  of  each  column.  The  width 
of  the  side  of  the  column  will  be  represented  by  6,  as  shown 
in  Figs.  4  and  6,  and  t  will  represent  the  total  thickness  of 
metal  whose  width  is  b,  also  as  indicated  in  the  same  figures. 
If  P  represents  the  total  load  on  one  jaw  of  the  post,  usually 
one  half  the  total  load  carried  by  the  post  or  column, 
the  average  working  intensity  of  pressure  on  the  section  of 
metal  bt  may  be  written 

P                      I  f\ 

-  =  9000-340-. (i) 

The  thickness  t  of •  metal  is  usually  the  quantity  desired, 
and  eq.  (i)  gives 

90006     26* 

In  these  equations  P  should  be  taken  in  pounds,  with 
b,  t,  and  /  in  inches. 

Eq;  (2)  has  been  used  to  a  considerable  extent  in  the 
design  of  steel  railroad  bridges,  and  it  is  probably  as  reason^ 
able  and  safe  a  value  of  the  thickness  t  as  can  be  written 
with  the  experimental  data  and  experience  now  available. 
It  is  applicable  to  steel  with  ultimate  tensile  resistance 
running  from  60,000  to  68,000  pounds  per  square  inch. 
For  higher  steel  or  for  highway  bridges,  or  for  other  struc- 
tures where  less  margin  of  safety  may  be  justifiable,  the 
value  of  t  may  be  made  correspondingly  less  than  that- 
given  in  eq.  (a);'_ ^_ 

Prob.  i.  It  is  required  to  design  a  mild-steel  pin-end 
column  45  feet  long  between  centres  of  pins  to  carry  a  load 
of  353,000  pounds.  The  column  formula  to  be  used  is 
essentially  that  given  as  eq.  (n)  of  Art.  83 : 

£  =  16,000-70-.  .....     .'    (^y 


c 


LLJ 


-4--M 


— D 


512  LONG   COLUMNS.  [Ch.  X. 

This  equation  gives  the  greatest  mean  intensity  allowed 
<«n  the  column,  so  that  p  multiplied  by  the  area  of  cross- 
section    to    be    determined    must    be 
^  equal   or  nearly   equal  to   232,000.   ? 

The  least  diameter  or  width  of  a 
built  column  should  not  exceed  about 
one  thirty-fifth  of  its  length,   except 
where  posts  or  columns  are  used  as 
-^    lateral  members,  when  the  length  may 
FlG  g  reach  as  much  as  40  times  the  least 

diameter    or    width    of    cross-section. 

In  this  case  the  column  is  to  be  built  of  two  plates  and 
four  angles,  as  shown  in  Fig.  8,  and  the  width  of  plate 
FG  must,  therefore,  not  be  less  than  about  16  inches.  A 
width  of  1 8  inches  will  make  a  well-proportioned  column 
and  that  dimension  will  be  assumed.  The  separation  of 
the  plates  is  preferably  made  such  that  the  moment  of 
inertia  of  the  section  about  the  axis  A  B  will  be  a  little  larger 
than  the  moment  about  the  axis  CD.  The  pin  will  pierce 
the  two  plates  so  that  its  axis  will  be  parallel  to  CD.  Under 
these  conditions,  if  the  column  is  designed  so  as  to  be  strong 
enough  with  the  moment  of  inertia  of  section  taken  about 
CD,  it  will  be  still  stronger  in  reference  tp  the  axis  AB,  and 
no  further  attention  need  be  given  to  possible  failure  about 
the  latter  axis. 

If  columns  of  this  type  are  proportioned  in  the  general 
manner  indicated,  the  radius  of  gyration  of  the  section 
about  the  axis  CD  will  be  approximately  .35  of  the  width. 
In  this  case  that  trial  radius  will,  therefore,  equal  6.3 
inches.  Hence,  inserting  the  values  of  £  =  540  inches  and 
r=6.3  inches  in  eq.  (3),  there  will  result  p  =  10,000  pounds 
per  square  inch.  The  total  area  of  section  required,  there- 
fore, will  be  closely  353,000  -=-10,000=35.3  sq.  ins.  The 
distribution  of  this  metal  between  the  plates  and  angles  is 


Art.  84.]     COMPLETE  DESIGN  OF   PIN-END  STEEL  COLUMNS.        513 

largely  a  matter  of  judgment.     Let  there  be  assumed 

Two    i8"Xf"  plates =22 . 5  sq.  ins. 

Four  3i"X  3i"X  1 1 -pound  angles =13      "     " 

Total =35  •  5  sq.  ins. 

This  is  a  tentative  composition  of  section  which  must  be 
tested  by  eq.  (3)  to  determine  whether  it  is  as  nearly 
accurate  as  it  should  be.  In  order  to  do  this,  the  moments 
of  inertia  of  the  section,  as  indicated,  must  be  taken  about 
the  two  axes  AB  and  CD. 

MOMENT  OP  INERTIA  ABOUT  CD: 

Two     i8"Xf"  plates =2X1x3*=  607.50 

Four   3^" X  3?"X  i  i-lb  angles  about  own  axis =  14. 20 

Four   3i"X3|"Xii-lb.anglesabout  CZ?=4X3-25X(7.99)2  =  829.92 

Moment  of  inertia =  1451 . 62 

MOMENT  OF  INERTIA  ABOUT  AB: 
Two     i8"X f "  plates  about  own  axis 2X1          —          . 74 

Two     1 8"  X  f  "plates  about  A  B 2X  n.25X(6.o6)2=   758.70 

Four   3^"X3£"X  n-lb.  angles  about  own  axis =     14.20 

Four   3i"X3i"Xn-lb.anglesabout,4£=4X3.25X(7.38)2  =   708.38 

Moment  of  inertia =  1482 .02 

These  computations  show,  first,  that  the  moment  of 
inertia  about  A  B  is  a  little  larger  than  that  about  CD, 
which  is  as  it  should  be.  They  also  show  that  the  radius  of 
gyration  r  is  6.39  inches.  The  approximate  rule  gives  r  = 
6.3  inches.  These  two  values  are  sufficiently  near  to  accept 
the  former.  The  trial  composition  of  section  may,  there- 
fore, be  considered  satisfactory  and  final.  The  thickness 
of  the  side  plates,  .625  inch,  is  sufficient  to  insure  no  buckling 
in  the  unsupported  width  between  rivets.  Similarly  the 
length  of  leg  of  the  3^-inch  angles  is  also  far  within  safe  or 
proper  limits.  All  features  of  the  cross-section  are,  there- 
fore, so  arranged  as  to  meet  all  the  requirements  of  suitable 
resistance  in  detail. 


514 


LONG    COLUMNS. 


[Ch.  X. 


The  details  of  the  ends  of  the  columns  where  they  are 
formed  into  jaws,  as  shown  by  Figs.  9  and  10,  still  remain 


o 

f 

o 

o 

o 

II 

0 

lo 

o 

0 

o 

0 

o 

6 

o 

>0 

o< 

/PI. 

.0 

o  V  o       o 

o. 

000 

X 

o     a,*5  o 

1 

y^> 

0 

0\   yt 

.     4- 

0 

000 

^inch 
batten  plate 


4 20': 


(3 


o 
o 
o 
o 
o 
o 

tt-> 

o 

o 


_i_. 


FIG.  9. 


FIG.  10. 


to  be  designed.  The  diameter  of  pin  will  be  taken  at  7 
inches,  as  shown  in  Fig.  9.  The  permissible  intensities 
of  shearing  and  of  the  bearing  on  the  walls  of  rivet  and  pin 
holes  will  be  taken  as  follows : 

Shearing  on  rivets   =  9000  pounds  per  sq.  in. 
Bearing  on  rivets  and  pins  =  16,000  pounds  per  sq.  in. 

The  total  thickness  of  metal  in  the  two  post  jaws  will, 
therefore,  be 

2  3 2OOO 

f        Thickness  of  metal  = -=2.1  inches. 

7  X loooo 

The  thickness  of  metal  in  each  jaw  must  therefore  be 
at  least  IT^  inches.  Inasmuch  as  the  thickness  of  side 
plates  of  the  column  is  f  inch,  the  pin  plates  to  be  riveted  to 
the  side  plates  must  be  at  least  TV  inch  thick  to  supply  the 


Art.  84.]     COMPLETE  DESIGN  OF  PIN-END  STEEL  COLUMNS.        515 

proper  bearing  surface  for  the  pin ;  but  that  thickness  must 
be  decided  by  the  formula  for  the  jaws,  eq.  (2).  In  that 
equation,  P  =  116,000  pounds,  while  b  =  iS  inches  and  /, 
from  Fig.  9,  is  9  inches.  Making  these  substitutions  in  eq, 

(2), 

/  =  i .  1 3  inches. 

In  order  to  meet  the  requirements  of  the  post-jaw  for- 
mula, therefore,  the  pin  plate  must  be  at  least  ^  inch 
thick.  It  is  essential  however  to  make  these  details 
specially  stiff  and  strong  and  the  thickness  will,  therefore, 
be  taken  at  -&  inch,  as  shown  in  Fig.  9. 

The  number  of  rivets  required  above  the  pin  hole 
would  ordinarily  be  computed  for  the  thickness  of  plate 
required  for  bearing  on  the  pin,  i.e.,  with  the  thickness  of 
pin  plate  of  j\  inch.  Assuming  that  thickness  for  this 
purpose,  the  rivets  being  taken  |  inch  in  diameter,  the 
bearing  value  of  a  single  rivet  will  be 

|XiVXi6, 000  =  6125  Ibs. 

The  single  shear  of  one  f-inch  rivet  at  9000  pounds  per 
square  inch  has,  a  value  of  5412  pounds  which  is  less  than 
the  bearing  value;  the  shear  will,  therefore,  decide  the 
number  of  rivets  required.  The  bearing  value  of  the  f-inch 
side  plate  on  the  pin  is  7X1X16,000  =  70,000  pounds. 
Hence  the  number  of  rivets  required  in  the  pin  plate  on 
each  side  of  the  column  will  be 

116000  —  70000 

' =  nine  rivets  (nearly). 

54i2 

These  nine  rivets  must  be  found  above  the  pin.  That 
number,  however,  is  far  too  small  for  the  pin  plate  acting 
as  a  part  of  the  jaw,  and  it  will  be  judicious  to  make  the 
total  number  of  rivets  above  the  pin  12,  as  shown  in  Fig.  9. 


516  LONC    COLUMNS.  [Ch.  X. 

The  jaw  plates  will  extend  5  inches  beyond  the  pin,  as 
shown.  The  two  batten  plates  above  which  the  latticing 
begins  will  each  be  taken  J  inch  thick,  and  they  will  be 
placed  as  shown  in  both  Figs.  9  and  10. 

It  is  assumed  that  the  ends  of  the  column  are  to  fit 
into  or  between  other  members  of  the  truss,  so  as  to  require 
cutting  away  the  legs  of  the  steel  angles,  as  shown,  as  this 
is  a  common  requirement. 

The  length  of  a  batten  plate  should  not  be  less  than 
its  width.  In  the  present  instance  the  width  of  batten  will 
be  19.75  inches;  the  length  will,  therefore,  be  taken  as 
20  inches. 

As  indicated  in  the  tabular  statement  at  the  beginning 
of  this  article,  the  lattice  bars,  fully  shown  in  Fig.  10,  will 
be  2^Xf  inches,  and  the  latticing  will  be  taken  as  double, 
although  this  is  not  always  done  for  the  size  of  column  in 
this  particular  instance.  The  lattice  bars  will  be  riveted 
at  their  intersections  also  as  shown  in  Fig.  10.  The  length 
of  lattice  bar  between  rivets  will  be  about  u  inches,  as 
the  angle  made  by  each  lattice  bar  with  the  side  of  the 
column  will  be  about  45  degrees.  A  single  f-inch  rivet, 
therefore,  at  the  end  of  each  bar  will  be  sufficient,  as  shown 
by  the  second  table  of  this  article.  At  each  panel  point 
of  latticing  a  single  J-inch  rivet  will  hold  the  ends  of  both 
lattice  bars. 

The  complete  bill  of  material  for  the  entire  column  will 
be  as  follows: 

Four   3i"X3i"Xii-lb.  angles,  46.42  ft.  long.  .185.7X11=2,043^3. 

Two    1 8"Xf"  plates,  46.42  ft.  long 93X38.25  =  3,557    " 

Four   27"X  ii"XTy'  Plates 9X21  =    189   " 

Four   2o"X 20" Xi"  battens 6|X34=    227    " 

24olin.  ft.  of  2$"Xf"  latticing 240X3.19=    766    " 

1060  |"  rivets 10.6X54=    572   " 


Total  weight  of  one  column =  7,354  Ibs. 


Art.  84.]     COMPLETE  DESIGN  OF  PIN-END  STEEL  COLUMNS.         517 


Prob.  2.  Let  it  be  required  to  design  a  mild-steel 
column  with  pin  ends,  36  feet  long  between  centres  of  pins, 
to  carry  a  load  of  225,500  pounds.  It  is  supposed  that 
the  column  is  a  member  of  a  railroad  bridge,  so  that  the 
load  given  includes  a  full  allowance  for  impact.  Gordon's 
formula  as  formerly  employed  in  the  American  Bridge  Com- 
pany's specification  will  be  used : 

17000 


nooor 

In  this  formula  p  is  the  greatest  mean  intensity  of 
working  pressure  allowed  on  the  section  of  the  column,  /  the 
length  between  centres  of  pins  in  inches, 
and   r  the   radius   of   gyration   of  the  j 

column  section  in  inches.  As  the  length 
of  the  column  is  but  36  ft.  =432  inches 
two  rolled  15 -inch  channels  latticed 
may  be  taken  as  the  principal  parts,  as 
shown  in  Fig.  n.  By  turning  to  the 
tables  in  any  steel  handbook,  it  will  be 
found  that  the  radius  of  gyration  of  a 
1 5 -inch  channel  about  the  axis  AB  varies  from  about  5.6 
inches  to  nearly  5.2  inches.  The  larger  of  the  two  values 
will  be  tentatively  employed.  Substituting  1  =  432  and 
r  =  5.6  in  the  above  formula  for  p, 

p  =  n,ooo  pounds  per  sq.  in. 
Hence  the  total  area  required  is 


*3  *-l~f- +- * 
j 
j 

P 

FIG.  ii. 


225500 

1 1 000 


=  20.5  sq.  ins. 


The_table  of  steel  channels  in  any  handbook  shows 
that  the  combined  area  of  two  15 -inch  3  5 -pound  channels 
is  20.58  sq.  in.,  and  they  will  be  accepted  as  correct.  The 


LONG  COLUMNS. 


[Ch.  X. 


same  table  gives  the  radius  of .  gyration  r  about  the  axis 
•AB,.  Fig.  H,  as  5.57  inches,  which  is  essentially  equal  to  the 
trial  value  5.6  inches. 

As  shown  in  Prob.  i,  it  is  desirable  to  have  the  moment 
of  inertia  of  the  section  about  A B,  Fig.  n,  a  little  less  than 
that  about  CD,  the  former  (A B)  being  parallel  to  the  axis 
of  the  pin.  Let  the  separation  of  the  channels  be  made 
10  inches  in  the  clear.  By  using  the  values  of  the  table, 
the  moments  of  inertia  about  the  two  axes  may  be- written: 

ABOUT  Axis  A  B: 
Moment  of  inertia  =320  X  2  =640. 

,      640 
HenCC  T  =2-0^8  =3I'°2;    •••—5.57  ms. 

ABOUT  Axis  CD: 
Moment  of  two  channel  sections  each  about  axis  parallel  to 

CD  and  through  centre  of  gravity 2X 8 . 48  =    16 . 96 


2X10.29X5-79 


=689.84 


Moment  of  inertia =  706 . 80 


O  ^O        O        O 

o     o     o     o 


'"" 


0       0    1   0       0 


_L 


yy  \ 

&f--.-WK'L-\ 

^> 

0 

o 

0 

o 

0 

0 

--> 

0 

o 

0 
0 
0 

+ 
1 

i 
r 

1 

V    ' 

! 

0 

,  FIG.  12.  FIG.  13. 

These  results  are  all  satisfactory  and  show  that  no 
revision  of  the  section  as  given  in  Fig.  1 1  is  needed. 

The  end  details  and  latticing  shown  in  Figs.  12  and  13 


Art.  84.]     COMPLETE  DESIGN  OF  PIN-END  STEEL  COLUMNS.        519 

remain    to    be    considered.     The    following    data    will    be 
required  : 

Thickness  of  channel  web  .........................  =  .43  inch. 

Allowed  shearing  on  rivets  and  pins  ................  =  io,cco  Ibs.  per  sq.  in. 

Allowed  bearing  on  rivets  and  pins  .................  =  20,000  Ibs.  per  sq.  in. 

Diameter  of  rivets  ...............................  =  f  inch. 

Diameter  of  pin  .................................  =6  inches. 

Value  of  one  f  -inch  rivet  in  single  shear  .............  =6,013  IDS- 

Bearing  of  pin  on  channel  web  ....................  =6X  .43  X  20,000 

=  5  1,  600  Ibs. 

Bearing  to  be  carried  by  pin  plate  =  —  —  —  •  —51,600=61,150  Ibs. 


Thickness  of  pin  plate  .................  ^ 

Bearing  value  of  one  £-inch  rivet  on  £-inch  plate  = 

|XiX20,ooo  =   8,750  Ibs. 

Hence  one  pin  plate  needs  =  ten  %-inch  rivets. 

It  is  assumed  that  the  ends  of  the  column  must  be 
formed  into  the  jaws  shown  in  Figs.  12  and  13.  As  indi- 
cated in  Fig.  12  the  mean  or  effective  length  of  the  jaw  is 
12  inches.  The  load  carried  by  one  jaw  is  112,750  pounds; 
hence  the  thickness  of  that  jaw  is  by  eq.  (2) 

-  =  l  A  inch  (nearfy)- 


8000x15 

The  thickness  of  the  jaw  or  pin  plate  to  be  riveted  to 
the  jaw  must  therefore  be  iTV—  •  43=11  inch.  In  order 
that  these  plates  may  be  firmly  made  a  solid  extension  of 
the  post  or  column  they  should  be  riveted  to  the  webs  of 
the  channels  with  the  rivets  shown  in  Fig.  12.  The  proper 
design  of  the  jaw,  therefore,  requires  a  much  longer  and 
thicker  plate  and  more  rivets  than  the  simple  consideration 
of  the  pin  and  rivet  bearing  and  shearing, 

The  width  of  channel  flange  is  3.43  inches,  hence  the 
total  width  of  column  over  these  flanges,  as  shown  in  Fig. 


520  LONG   COLUMNS.  [Ch.  X. 

13,  is  i6|  inches.  Each  batten  plate  is  therefore  taken  as 
17  inches  by  18  inches. 

The  length  of  each  lattice  bar  of  the  single,  3o-degree 
latticing  will  be  about  16  inches  between  centres  of  rivets 
at  their  ends.  Lattice  bars  2\  inches  by  f  inch  in  section 
will,  therefore,  be  used. 

The  complete  bill  of  material  for  one  column  will  then  be 

Two  15"  35-lb.  channels  37^  ft.  long 2X  35 X  3?i  =  2,602  Ibs. 

Four         1 3" X 30" X \\ "plates 10X41. 44     =    415    " 

Four         1 7"  X  1 8" X£"  plates 6X28.9       =    173    " 

Forty-six  2j"Xf"X  19"  bars 72X   3- 19     =    230    " 

Two  hundred  and  twenty-five  \"  rivets 2^X54  =     J22    " 

Total  weight  of  one  column =  3,542  Ibs. 

Art.  85. — Cast-iron  Columns. 

Cast  iron  was  the  earliest  form  in  which  the  metal 
iron  was  used  for  'columns,  and  it  is  natural,  therefore, 
that  the  first  long-column  formulae  for  cast  iron  should  have 
been  among  the  earliest  for  that  class  of  members.  The 
first  experimenter  was  Eton  Hodgkinson,  who  published 
the  results  of  his  tests  on  small  cast-iron  columns,  the 
greatest  length  of  which  was  but  60.5  inches,  in  the  "  Philo- 
sophical Transactions  of  the  Royal  Society  of  London  for 
1840."  He  not  only  recognized  the  round-  and  fixed-end 
conditions,  but  he  also  made  the  distinction  between  long 
columns  and  short  blocks,  the  length  of  the  latter  being 
from  4  to  5  times  the  diameter  or  least  cross-section  dimen- 
sion. If  d  be  the  diameter  of  the  column  in  inches  and  / 
the  length  in  feet,  and  in  the  case  of  hollow  round  columns 
if  D  be  the  exterior  diameter  in  inches  and  d  the  interior 
diameter  in  the  same  unit,  while  P  is  the  total  or  ultimate 
load  in  pounds  on  the  column,  Hodgkinson  established 
the  following  formulae  for  long  cast-iron  columns: 

^3.76 
P  =  33.379777- ;  (for  rounded  ends).      .     .     .     (i) 


Art.  8s.]  CAST-IRON  COLUMNS.  521 

ds-ss 
P  =  98,  922-7^-;  (for  fixed  ends)  .....     (2) 

For  hollow  cylindrical  columns  of  cast  iron 

£)3-  76  _  ^3-76 

P  =  29,120  --  77^  --  ;  (for  rounded  ends).     .     (3) 


P  =99,320  ---  Y-1  --  '    (f°r  fiXed  endS)'         '         •         (4) 

The  working  or  maximum  load  allowed  in  any  design 
of  cast-iron  columns  would  be  found  by  taking  one  fifth  to 
one  eighth  of  the  values  given  in  eqs.  (i)  to  (4)  inclusive. 
It  will  be  observed  that  Hodgkinson's  formulae  expressed 
in  the  preceding  equations  are  simply  Euler's  formulae 
as  given  in  eqs.  (6)  and  (9)  of  Art.  35.  with  the  introduction 
of  an  empirical  coefficient  and  with  the  indices  of  d  and  / 
changed  to  harmonize  with  the  experimental  results. 

As  Hodgkinson's  experiments  were  made  on  very 
small  columns  of  different  metal  from  that  used  in  cast- 
iron  columns  of  the  present  day,  his  formulae  cannot  safely 
be  used  for  practical  purposes  at  the  present  time. 

A  correct  formula  for  cast-iron  columns  must  be  based 
upon  tests  of  full-size  columns  cast  with  the  metal  ordi- 
narily employed  in  structural  practice.  Such  tests  have 
been  made  at  the  U.  S.  Arsenal  at  Watertown,  Mass.,  and 
will  be  found  reported  in  H.  R.  Ex.  Doc.  No.  45,  5oth  Con- 
gress, 2d  Session,  and  in  H.  R.  Ex.  Doc.  No.  16,  5oth 
Congress,  ist  Session.  A  valuable  series  of  tests  was  also 
made  at  Phcenixville,  Pa.,  at  the  works  of  the  Phoenix 
Bridge  Co.,  under  the  auspices  of  the  Department  of 
Buildings  of  New  York  City  in  1896-97.  Although  the 
entire  series,  including  both  the  tests  at  Watertown  and 
Phoenix  ville,  do  not  cover  the  variety  of  sectional  forms 
and  range  of  ratio  of  length  to  diameter  that  could  be 


522 


LONG  COLUMNS. 


[Ch.  X. 


H 
co 

h 


SI 

z  co 
<  ->' 


O  + 


f 
f'/ 


Art.  85.] 


CAST-IRON  COLUMNS. 


523 


desired,  the  results  are  sufficiently  extended  to  show  closely 
what  may  be  considered  the  proper  ultimate  values  for 
hollow  round  cast-iron  columns  of  full  size. 

TABLE  I. 


No. 

Length  in 
Inches. 

Diameter  in  Inches. 

Area  of 
Section 
in 
Square 
Inches. 

Length 
over 
Exterior 
Diameter 

Ultimate 
Resistance 
in  Pounds 
per  Square 
Inch. 

Large  End. 

Small  End. 

Ext. 

Int. 

Ext. 

Int. 

I 

190^.25 

15 

13 





43.98 

12.7 

30,830 

2 

15 

T    C 

12.75 

TO      1  C 





49-03 

A  C\     f*1 

12.7 

TO      *7 

27,126 

O  /I    A  1  A 

4 

' 

10 

*'*-7a 

12.75 



49  -Uv5 
49.48 

1^  .   / 
12.7 

^4,4v)4 
25,182 

5 

' 

15 

12.66 

50.91 

12.7 

35,435 

6 

' 

15 

12.63 

— 



5L52 

12.7 

40,41  1  * 

7 

160 

8 

6 





21.99 

20 

29,604 

8 

1  60 

8 

5-91 





22.87 

20 

28,229 

9 

1  20 

6.06 

3-78 





17.64 

2O 

25,805 

10 

1  20 

6.09 

3-96 





17-37 

2O 

26,205 

ii 

147-75 

8 

6-5 

17.08 

I8.5 

25,973 

12 

150 

9 

7 





25.14 

I6.7 

21,183 

13 

162 

12 

10 





34-55 

13-5 

30,813 

14 

159-75 

14 

12 





40.84 

ii.  4 

25,400 

15 

169 

5 

4-54 

3-5 

34 

29,854 

16 

157 

7.17 

4.83 



21.8 

22 

25,470 

17 

157 

6.35 

3-9 

•  

17.28 

25 

27,210 

18 

156 

5-8 

4-03 



13.22 

27 

25,100 

19 

142.6 

7.68 

5-52 

5-94 

4-3 

17-49 

26.7 

29,310 

20 

146.8 

8.01 

5.58 

5-9 

4-35 

18.65 

21.3 

28,520 

21 

150 

6.17 

4-85 

5-09 

3-48 

12.08 

27 

33,500 

22 

145-5 

6 

4-35 

4-74 

2-73 

12.  8l 

37-i 

24,620 

23 

133-6 

6.  02 

4.36 

4.84 

2.88 

12.87 

24.6 

28,060 

24 

129.3 

6.03 

4-35 

4.87 

2-95 

12.87 

23-7 

27,350' 

25 

127.6 

7-47 

5-97 

5.72 

4.62 

12.13 

19-3 

46,660 

26 

118.5 

3-98 

i  .96 

2.97 

1.49 

7.16 

34-1 

23,090 

27 

119 

3.98 

1.96 

2.98 

i-47 

7.17 

34-3 

22,040 

28 

118 

3-97 

1-95 

2-99 

1-39 

7.1" 

34-2 

25,060 

29 

84.6 

4.88 

3-03 

4-27 

2.08 

18.5 

31,190 

*  Not  broken. 

Table  I  shows  the  results  of  all  these  tests,  while  the 
Plate  exhibits  the  same  results  graphically.  The  tests 
Nos.  i  to  io- inclusive  were  made  at  Phcenixville  in  De- 
cember, 1897,  and  Nos.  ii  to  14  inclusive  in  1896;  the 


524  LONG  COLUMNS.  [Ch.  X. 

former  group  under  the  immediate  direction  of  Mr.  W.  W. 
Ewing,  and  the  latter  under  the  immediate  direction 
of  Mr.  Gus  C.  Henning.  The  results  shown  for  tests  15  to 
1 8  inclusive  were  taken  from  H.  R.  Ex.  Doc.  No.  45,  5oth 
Congress,  26.  Session,  but  those  for  Nos.  19  to  29  inclusive 
are  either  taken  or  digested  from  H.  R.  Ex.  Doc.  No.  16, 
5oth  Congress,  ist  Session,  being  portions  of  reports  of 
tests  of  metals  and  other  materials  at  the  United  States 
Arsenal,  Watertown,  Mass. 

As  'fable  I  shows,  the  columns  Nos.  19  to  29  inclusive 
were  slightly  conical,  although  probably  not  enough  so  to 
affect  appreciably  their  resistances.  The  areas  of  section 
in  square  inches  for  these  columns  were  taken  at  mid- 
distance  between  their  ends.  As  the  area  of  section  varied 
considerably  in  some  columns  that  operation  may  be  a 
source  of  a  little  error  in  determining  the  ultimate  resist- 
ance per  square  inch  from  the  result  of  the  tests,  but  if  the 
error  exists  at  all  it  must  be  very  small.  The  mid-external 
diameter  was  also  taken  for  these  columns  in  determining 
the  ratio  of  the  length  over  the  diameter  shown  in  the 
Table  and  in  the  Plate. 

As  will  be  observed  both  in  the  Table  and  in  the  Plate, 
the  ultimate  resistances  per  square  inch  determined  by 
the  tests  are  quite  variable,  even  for  the  same  ratio  of 
length  over  diameter.  Indeed,  in  a  number  of  cases  they 
are  quite  erratic.  In  Nos.  i  to  6,  for  which  the  ratio  of 
length  over  diameter  was  12.7,  the  ultimate  resistances 
vary  from  a  little  over  24,000  Ibs.  per  square  inch  to  over 
40,000  Ibs.  per  square  inch  with  no  failure  at  the  latter 
value.  Again,  the  ultimate  resistance  per  square  inch, 
for  No.  25,  which  shows  a  ratio  of  length  over  diameter  of 
less  than  20,  is  nearly  47,000  Ibs.  per  square  inch,  which  is 
excessively  high  as  compared  with  other  ultimate  resist- 
ances with  the  same  or  less  ratio  of  length  over  diameter. 


Art.  85.]  CAST-IRON  COLUMNS.  525 

These  erratic  results  are  not  surprising  in  view  of  the 
ordinary  character  of  the  metal.  It  should  be  remembered 
that  the  failures  of  these  columns  are  frequently  recorded 
with  such  ' '  remarks ' '  as  the  following :  ' '  Foundry  dirt  or 
honey-comb  between  inner  and  outer  surfaces,"  "bad 
spots,"  "cinder  pockets  and  blow  holes  near  middle  of 
column,"  "flaws  and  foundry  dirt  at  point  of  break." 
In  other  words,  it  was  no  uncommon  feature  to  observe  that 
defects,  flaws,  or  blow  holes  or  thin  metal  had  determined 
the  place  of  failure.  There  is  considerable  uncertainty  in 
platting  the  results  of  tests  affected  by  these  abnormal  con- 
ditions, but  a  more  or  less  satisfactory  law  for  the  generality 
of  cases  may  be  determined  from  a  graphical  representation 
of  the  results,  as  shown  on  Plate  I.  On  that  Plate  the 
ultimate  resistances  in  pounds  per  square  inch,  as  shown 
in  Table  I,  have  been  platted  as  vertical  ordinates,  while 
the  ratios  of  length  over  diameter  given  in  the  same  Table 
are  represented  by  the  horizontal  abscissas,  all  as  clearly 
shown.  The  full  straight  line  drawn  in  about  a  mean 
position  among  the  results  of  the  tests  probably  represents 
as  near  as  any  that  can  be  found  a  reasonable  law  of  variation 
of  ultimate  resistance  with  the  ratio  of  length  over  diameter. 
It  is  evident  that  within  the  range  of  these  experiments  a 
straight  line  will  represent  the  ultimate  resistances  fully 
as  well  as  any  curve,  if  not  better,  although  the  results  for 
the  lengths  of  thirty-four  times  the  diameter  begin  to 
indicate  a  little  curvature.  The  formula  which  represents 
this  straight  line,  i.e.,  which  gives  the  ultimate  resistance 
per  square  inch,  is  as  follows: 

p  =  30,500  -160^ (5) 

It  is  to  be  borne  in  mind  that  these  columns  were  round 
and  hollow,  and  that  they  were  tested  with  flat  ends  in  all 


526  LONG  COLUMNS.  [Ch.  X. 

cases.  The  ordinary  formula,  based  upon  Hodgkinson's 
tests,  and  frequently  used  in  cast-iron  column  construction, 
is  as  follows: 

80000 

t=    — ? (v 

400  d2 

The  curve  corresponding  to  this  particular  form  of 
Tredgold's  formula  is  also  shown  on  the  Plate.  It  will  be 
seen  that  at  the  ratio  of  length  over  diameter  of  10  to  12 
(not  an  uncommon  ratio)  the  ultimate,  as  given  by  this 
formula,  is  just  about  double  that  shown  by  actual  test.  In 
other  words,  if  a  safety  factor  of  5  were  required,  as  is  the  case 
in  some  building  laws,  the  actual  safety  factor  would  be  but 
2^.  The  curve  represented  by  eq.  (6)  is  seen  to  cross  the  true 
curve  at  a  ratio  of  length  over  diameter  of  about  29.  A 
glance  at  the  Plate  will  show  how  erroneous  and  dangerous 
is  the  use  of  the  usual  formula  for  hollow  round  cast-iron 
columns;  indeed,  that  formula  is  grossly  wrong,  both  as  to 
the  law  of  variation  and  the  values  of  ultimate  resistance. 

In  view  of  the  working  resistances,  which  have  been 
used  in  the  design  of  cast-iron  columns,  it  is  no  less  interest- 
ing than  important  to  compare  the  ultimate  resistances  per 
square  inch  of  mild-steel  columns,  as  determined  by  actual 
tests,  with  the  ultimate  resistances  of  cast-iron  columns, 
as  shown  by  the  tests  under  consideration.  The  broken 
line  of  short  dashes  represents  the  formula 

£  =  52,000-180- (7) 

a 

determined  by  actual  tests  of  mild-steel  angles  made  by 
Mr.  James  Christie  at  the  Pencoyd  Bridge  Works,  and 
given  in  Art.  60.  This  line  or  formula  shows  that  the 
ultimate  resistances  per  square  inch  of  mild-steel  columns 


Art.  85.]  CAST-IRON  COLUMNS.  527 

are  from  40  to  50%  greater  than  the  corresponding  quanti- 
ties for  cast-iron,  the  same  ratio  of  length  over  diameter 
being  taken  in  each  comparison. 

When  the  erratic  and  unreliable  character  of  cast-iron 
columns  is  considered,  it  is  no  material  exaggeration  to 
state  that  these  tests  show  that  the  working  resistance 
per  square  inch  may  be  taken  twice  as  great  for  mild-steel 
columns  as  for  cast-iron;  indeed,  this  may  be  put  as  a 
reasonably  accurate  statement. 

The  series  of. tests  of  cast-iron  columns  represented  in 
the  Plate  constitute  a  revelation  of  a  not  very  assuring 
character  in  reference  to  cast-iron  columns  now  standing, 
and  which  may  be  loaded  approximately  up  to  specification 
amounts.  They  further  show  that  if  cast-iron  columns 
are  designed  with  anything  like  a  reasonable  and  real  margin 
of  safety  the  amount  of  metal  required  dissipates  any 
supposed  economy  over  columns  of  mild  steel. 

If  the  average  working  stress  per  square  inch  is  one 
fourth  of  the  ultimate  resistance,  eq.  (5)  gives 

I 
£  =  7600-40^ (8) 

If  the  working  stress  is  to  be  taken  at  one  fifth  the 
ultimate,  eq.  (5)  gives 

£  =  6100-32^ (9) 

In  these  equations  p  is  the  average  working  intensity 
of  pressure  in  pounds  per  square  inch.  The  length  /  and  the 
exterior  diameter  d  must  be  taken  both  in  the  same  unit, 
ordinarily  the  inch. 

These  formulae  may  be  used  between  the  limits  of  -  =  10 

d 

and  -  =  35  or  even  40.     They  may  also  be  applied  to  hollow 


528  LONG  COLUMNS.  [Ch.  X. 

rectangular  columns  with  reasonably  close  approximation, 
d  being  taken  as  the  smaller  exterior  side  of  the  rectangular 
cross-section. 

Art.  86. — Timber  Columns. 

The  greater  part  of  available  tests  of  full-size  timber 
columns  have  been  made  prior  to  1900,  and  their  results 
have  not  been  obtained  either  by  the  aid  of  improved  appli- 
ances in  testing  now  employed,  or  in  all  respects  under  the 
care  given  in  later  testing  work  to  secure  accuracy  or  to 
avoid  misinterpretation  of  the  more  or  less  obscure  condi- 
tions which  attend  the  testing  of  full-size  timber  members. 

The  ratio  of  the  length  divided  by  the  radius  of  gyration 
is  much  less  in  timber  columns  than  those  of  iron  or  steel. 
Furthermore,  as  sections  taken  at  right  angles  to  the  axes 
of  timber  columns  are  almost  always  rectangular,  it  is  per- 
missible to  use  the  ratio  of  the  length  over  the  least  side 
rather  than  the  length  over  the  least  radius  of  gyration, 
gaining  thereby  a  little  simplicity  in  the  use  of  column 
formulae. 

Timber  columns  are  subject  to  the  same  vicissitudes  of 
knots,  wind-shakes,  season  cracks  and  decay  as  other  timber 
members.  Indeed  most  failures  of  full-size  timber  mem- 
bers are  induced  by  some  local  defect  such  as  a  knot,  either 
decayed  or  sound.  Unless  in  a  thoroughly  protected  place, 
timber  columns  are  in  a  condition  of  almost  constant  change 
and  in  the  long  run  for  the  worse. 

The  degree  of  seasoning  is  an  element  of  material  effect 
in  the  resistance  of  timber  columns.  The  greater  the 
amount  of  moisture  in  timber,  the  less  will  be  its  capacity 
for  compressive  resistance,  other  conditions  remaining  un- 
changed. As  in  all  other  full-size  timber  tests,  the  con- 
dition of  moisture  should  be  known  and  stated  in  connection 
with  the  results  of  timber  column  tests.  It  makes  little 


Art.  86.] 


TIMBER  COLUMNS. 


529 


or  no  difference  whether  the  moisture  is  the  original  sap  or 
the  result  of  a  damp  atmosphere  or  immersion  in  water. 

Among  the  earliest  tests  were  those  of  Professor  Lanza, 
who  investigated  timber  mill  columns,  mostly  of  circular 
section  and  some  of  them  after  standing  in  use  in  com- 
pleted buildings  for  various  periods  from  one  year  to  twenty- 
five  years.  These  columns  varied  in  length  from  about  2 
to  14  feet,  the  great  majority  of  them  being  from  n  to  14 
feet.  The  diameters  varied  generally  from  about  5  inches 
to  about  1 1  inches.  A  few  were  square.  Neither  the  shape 
nor  the  dimensions  of  cross-sections  appeared  to  affect 
materially  the  results  of  tests.  The  principal  results  of 
these  tests  are  given  in  the  tabulated  statement  below: 


Max. 
Lbs.  per  Sq.  In. 

Mean. 
Lbs.  per  Sq.  In. 

Min. 
Lbs.  per  Sq.  In. 

Yellow  pine,  partially  seasoned 
Yellow  pine,  air  seasoned  
Yellow  pine,  dock  seasoned  .  .  . 
White  wood,  partially  seasoned 
White  oak,  partially  seasoned  . 
White  oak,  in  mill  6^  years  .  .  . 
White  oak,  in  mill  25  years  .  .  . 
White  oak,  thoroughly  seasoned 

5,450 
4,892 
5,950 

3,333 
3,786 
6,029 
6,147 
4,450 

4,370 
4,690 
4,563 

3,010 
3,070 
4,170 
4,420 
3,175 

3,510 

4,488 

3,477 
2,687 

1,964 
2,945 
3,266 
1,865 

The  ends  of  these  columns  were  usually  flat,  sometimes 
with  a  so-called  "  pintle  "  or,  in  a  few  cases,  one  end  round. 
These  results  show  the  usual  erratic  features  of  full-size 
timber  tests,  some  of  which  doubtless  are  due  to  undiscovered 
weaknesses  at  some  point.  Prof.  Lanza  stated  that  "  The 
immediate  location  of  the  fracture  was  generally  determined 
by  knots."  Some  of  the  columns  were  tapered  and  the 
reduction  of  the  section  at  the  ends  of  such  columns  usually 
located  the  failure  at  those  reduced  ends. 

The  greatest  ratio  of  length  to  radius  of  gyration  in 
these  columns  was  about  86,  but  the  actual  results  did  not 
show  that  there  was  any  discoverable  relation  between  the 


53° 


LONG  COLUMNS. 


[Ch.  X. 


ratio  of  the  length  over  the  radius  of  gyration  and  the 
ultimate  column  resistance.  The  latter  was  influenced 
little  or  none  by  the  length  of  the  columns. 

Tables  I  and  II  show  the  results  of  the  early  tests  of 
Col.  Laidley,  Engineer  Corps,  U.  S.  A.,  made  many  years 
ago  and  reported  in  "Ex.  Doc.  12,  47th  Congress,  ist 
Session."  They  show  the  large  increase  in  ultimate  resist- 
ance per  square  inch  with  short  lengths.  Indeed  some  of 
the  pieces  were  short  blocks.  These  results  indicate  the 
care  that  should  be  taken  in  discriminating  between  the 
ultimate  compressive  resistances  of  short  timber  blocks  and 
long  columns.  The  results  in  Table  I  for  those  pieces 
seasoned  twenty  "years  are  too  high,  while  those  for  pieces 
Nos.  16,  17,  and  18  are  low,  in  consequence  of  the  material 

TABLE  I. 
YELLOW  PINE. 


No. 

Length, 
Inches. 

Form  of  Section. 

Section  Dimensions, 
Inches. 

Ultimate  Resistance 
per  Sq.  In. 

Lbs. 

I 

20.4 

Circular. 

10.  2  diam. 

6,676^ 

2 

119-95 

Square. 

ii      Xn 

6,230 

1u 

3 

119.90 

ii      Xn 

6,552 

9 

4 

20.0 

10.4X10.4 

7,936 

w 

rt 

5  ^ 

16.0 

8X8 

8,165 

DJ 

Cfl 

6  * 

8.0 

4X4 

7,394 

'c    ' 

7 

3-0 

i.5X    i.5 

5,533 

OS    VH 

8 

6.0 

3     X    3 

8,644 

•l! 

9 

6.0 

3     X    3 

8,133 

la 

10 

3-0 

i-5X    i.5 

8,389 

oj  cs 

ii 

3-0 

i-5X    1.5 

8,302 

M 

12 

3-o 

i-5X    1.5 

6,355 

4 

13 

14.0 

4-6X   4-6 

9,947 

Dfl 

"c3 

14 

17.2 

4-6X   4-6 

10,250 

S 

15 

19.1 

5-3X    5-3 

7,820 

C/3 

16 

180.0 

Rectangular. 

16     XI3-65 

3,070 

17 

180.0 

' 

i6.2X    7-0 

2,795 

18 

180.0 

17     X    8.75 

3,180 

Nos.  13,  14,  and  15  were  pine  of  very  slow  growth. 
Nos.  1 6,  17,  and  18  were  very  green  and  wet. 


Art.  86.] 


TIMBER  COLUMNS. 


531 


TABLE  II. 

SPRUCE  THOROUGHLY  SEASONED. 


No. 

Length, 
Inches. 

Form  of  Section. 

Section  Dimensions, 
Inches. 

Ultimate  Resistance 
per  Sq.  In. 

Lbs. 

I 

24 

Rectangular. 

5.4X5.4 

4,946 

2 

'  24 

5.4X5.4 

4,811 

3 

36 

5.4X5.4 

4,874 

4 

36 

5.4X5.4 

4,500 

5 

60 

5.4X6.4 

4,451 

6 

60 

5-4X6.4 

4,943 

7 

1  2O 

5.4X5.4 

3,967 

8 

1  2O 

5.4X5.4 

4,908 

9 

60 

5.4X5.4 

5,275 

10 

30 

5.4X5.4 

5,372 

IT 

15 

5,4X5-4 

5,754 

12 

121  .  2 

Circular. 

12.4  diam. 

4,681 

being  green  and  wet.  The  tests  pieces  in  Tables  I  and  II 
were  generally  fine  straight-grained  timber  of  better  quality 
than  ordinarily  used  in  engineering  practice. 

This  condition  accounts  largely  for  Col.  Laidley's  results, 
being  materially  higher  than  Prof.  Lanza's  for  the  same 
kind  of  timber. 


Formula  of  C,  Skater  Smith. 

Although  these  formulae  were  deduced  from  tests  made 
many  years  ago,  they  have  been  so  extensively  used  over 
such  a  long  period  that  they  may  properly  be  considered 
among  the  classics  of  engineering  literature  of  this  kind, 
Hence  they  are  given  here,  although  not  now  used  so 
widely  as  formerly. 

The  tests  of  full-size  sticks  on  which  the  formulae  are 
based  were  grouped  by  Mr.  Smith  as  indicated  and  the 
corresponding  formulae  are  as  given  below. 


532  LONG  COLUMNS.  [Ch.  X. 

"  i  st.  Green,  half  -seasoned  sticks  answering  to  the 
specification  'good,  merchantable  lumber.' 

"  2d.  Selected  sticks  reasonably  straight  and  air-sea- 
soned tinder  cover  for  two  years  and  over. 

"3d.  Average  sticks  cut  from  lumber  which  had  been 
in  open-air  service  for  four  years  and  over." 

If  /  =  length  of  column  in  inches, 

d  =  least  side  of  column  section  in  inches, 
and          p  =  Ult.  Comp.  resistance  in  Ibs.  per  sq.  in.  ; 

then  the  formulae  found  for  these  three  groups  were  : 
>orNo.i:/>=     54°°,2. 


_      __  8200 

For  No.  2  :  p  =  —  - 


For  No.  3:^=     5°°° 


But  in  order  to  provide  against  ordinary  deterioration 
while  in  use,  as  well  as  the  devices  of  unscrupulous  builders, 
Mr.  Smith  recommends  the  formula  for  group  No.  3  as  the 
proper  one  for  general  application.  He  also  recommended 

that   the   factor  of  safety  be  v/—  until    25   diameters  are 

\d 

reached,  and  five  thenceforward  up  to  60  diameters. 
This  last  limit  he  regards  as  the  extreme  for  good 
practice. 


Art.  86.] 


TIMBER  COLUMNS. 


533 


Tests  of  White  Pine  and  Yellow  Pine  Full-size  Sticks  with 

Flat  Ends. 

In  consequence  of  the  usual  manner  of  simply  abutting 
the  end  of  timber  columns  against  their  supports,  all  such 
members  are  practically  always  assumed  to  have  flat  ends, 
but  this  expression  does  not  mean  accurately  squared  "  flat 
ends."  Tables  III  and  IV  have  been  formed  by  digesting 
the  results  of  tests  of  nearly  or  quite  full-size  white  and 
yellow  pine  timber  columns  made  at  the  U.  S.  Arsenal  at 
Watertown,  Mass.,  and  reported  in  "  Ex.  Doc.  No.  i,  47th 
Congress,  26.  Session,"  constituting  one  of  the  best  series 
of  timber  column  tests  yet  made  in  this  country. 

Each  result  in  both  Tables  is  usually  a  mean  of  from 
two  to  four  tests,  although  a  few  belong  to  one  test  only. 
All  timber,  both  of  yellow  and  white  pine,  was  ordinary 
merchantable  material,  with  about  the  usual  defects,  knots, 
etc.,  and  failure  frequently  took  place  at  the  latter ;  it  was  all 
well  seasoned,  and  all  columns  were  tested  with  flat  ends. 

TABLE  III. 
YELLOW-PINE  COLUMNS  WITH  FLAT  ENDS. 


Ultimate 

Ultimate 

Length. 

Size  of  Stick, 
Inches. 

/ 
d' 

Compres- 
sive  Re- 
sistance, 

Length. 

Size  of  Stick, 
Inches. 

/ 
T 

Compres- 
sive  Re- 
sistance, 

Lbs.  per 

Lbs.  per 

Sq.  In. 

Sq.  In. 

Ft.   Ins. 

Ft.  Ins. 

15       0 

8.25X16.25 

21.7 

3,445 

15       0 

5.0X12 

35-6 

3,764 

IO      O 

5-5   X    5-5 

22 

4,738 

23     4 

7-7X   9-7 

36.4 

3,304 

16     8 

7-7   X   9-7 

26.7 

4,384 

17     6 

5-5X    5-5 

38.2 

3,242 

15       0 

6.6   Xi5-6 

27.0 

3,593 

15     o 

4-5X11.6 

4i 

2,462 

12       6 

5-5  X   5-5 

27-3 

5,077 

26     8 

7-4X   9-4 

43 

2,893 

15      0 

5.9    XI2.0 

30-8 

3,546 

15     o 

4.oX  11.4 

44 

3,065 

20     o 

7-6  X   9-6 

31.2 

3,496 

20     o 

5-4X    5-4 

44-3 

2,867 

15       0 

5-7   Xii-7 

31-9 

3,106 

22       6 

5-5X    5-5 

50 

2,065 

15       0 

5-6  XI5-6 

32.1 

3,656 

25       0 

5-5X    5-5 

55 

I,856 

J5     o 

5-5   X    5-5 

32-8 

3,962 

27     6 

5-3X    5-3 

62.3 

1,709 

S3  4  LONG  COL  UMNS  [Ch.  X. 

Flat -end  yellow-pine  columns  were  observed  to  begin  to 
fail  with  deflection  at  a  length  of  about  22d,  d  being  the 
width  or  least  dimension  of-  the  normal  cross-section.  All 
columns  were  of  rectangular  section,  and  /  in  the  following 
table  is  the  length.  Table  III,  therefore,  includes  no  short 
column,  i.e.,  one  which  failed  by  compression  alone  with 
no  deflection. 

About  sixteen  of  the  latter  were  tested  with  the  follow- 
ing results : 

Short  yellow-pine  columns;  (  maximum  =  5,67 7  Ibs.  per  sq  in. 
/H-d  below  22.  Mmean          =  4,442 

(  minimum  =  3,430    ' 

Each  of  the  preceding  tests  was  made  on  a  single  rectan- 
gular stick.  A  number  of  tests,  however,  were  made  on 
compound  columns  formed  by  bolting  together  from  two 
to  three  rectangular  sticks,  with  bolts  and  packing  or 
separating  blocks  at  the  two  ends  and  at  the  centre.  The 
bolts  were  parallel  to  the  smaller  sectional  dimensions  of 
the  component  sticks.  As  was  to  be  expected,  those 
compound  columns  possessed  essentially  the  same  ultimate 
resistance  per  square  inch  as  each  component  stick  con- 
sidered as  a  column  by  itself,  as  the  following  results  show. 
/  is  the  length  of  the  column  and  d  the  smallest  dimension 
or  width  of  one  member  of  the  composite  column.  All 
had  flat  ends. 

l  +  d.         Number  of  Tests. 

{maximum -=4,5  59  Ibs.  per  sq.  in. 
mean          =3,841 
minimum  =2,756 
(maximum  =  3, 357 

36.... 18 -jmean          =3,122 

(minimum  =2,942 

Table  IV  gives  the  results  for  white-pine  columns,  and 
corresponds  with  Table  III,  in  that  it  shows  only  the  failures 
with  deflection,  which  was  observed  to  begin  with  those 
columns  at  a  length  of  32^.  /  and  d  possess  the  same 


Art.  86.] 


TIMBER  COLUMNS. 


535 


TABLE  IV. 

WHITE-PINE  COLUMNS  WITH  FLAT  ENDS. 


• 

Ultimate 

Ultimate 

Length. 

Size  of  Stick, 
Inches. 

/ 
~~j* 

Compres- 
sive  Re- 
sistance, 

Length. 

Size  of  Stick, 
Inches. 

I 

d  ' 

Oompres- 
sive  Re- 
sistance, 

a 

Lbs.  per 

Lbs.  per 

Sq.  In. 

Sq.  In. 

Ft.  Ins. 

Ft.  Ins. 

15       0 

5.6X15.6 

32 

1,874 

I7      6 

5-4X5.4 

40 

1,841 

20     3 

7-4X   9-3 

32.4 

2,448 

26     8 

7.5X9-3 

42.7 

2,H3 

15       0 

5-6X11.5 

32.7 

2,432 

20     o 

5.3X5.3 

45 

i,455 

15     3 

5-4X    5-4 

33 

2,744 

22       6 

5.2X5.2 

52 

1,501 

23     4 

7-7X   9-6 

36.4 

2,072 

25     o 

5-3X5-3 

57 

952 

15       0 

4-5X11.6 

40 

1,672 

27     6 

5.4X5.4 

62 

r  08  1 

signification    as   in   Table  III,   the    column   l  +  d   showing 
the  ratios  between  the  lengths  and  least  widths. 

Thirty  columns  with  lengths  less  than  32^  were  tested 
to  destruction.  These  sticks  failed  generally  at  knots  by 
direct  compression  and  without  deflection.  The  results 
of  these  thirty  tests  were  as  follows: 


Short  white-pine  columns ; 
l-r-d  below  32 


maximum 


minimum 


3,700  Ibs.  per  sq.  in. 
2,414  "   " 
1,687  "   " 


All  the  preceding  white-pine  columns  were  single  sticks, 
but  a  large  number  of  built  posts  composed  of  two  to  four 
white-pine  sticks  bolted  together,  with  spacing  blocks  at  the 
two  ends  and  at  the  centre,  were  also  tested  with  the  results 
shown  below.  /  -f-  d  is  the  ratio  of  length  over  least  width  of 
a  single  stick  of  the  set  forming  the  composite  column. 

l-±-d.  Number  of  Tests . 

(  maximum  =  2, 273  Ibs.  per  sq.  in. 


36. 


•  I  15... 

9-  ... 
6.  .. 

.  .  .  •<  mean          =  1,980 
(  minimum  =  1,661 
{maximum  =  2,  255 
mean          =1,999 
minimum  =1,804 
!  maximum  =  2,02  1 
mean          —  i  830 

minimum  =1,419 

536 


LONG  COLUMNS. 


[Ch.  X. 


A  comparison  of  these  results  with  those  given  in  Table 
IV  shows  that  these  composite  or  built  columns  were  the 
same  in  strength  per  square  inch  with  the  single  sticks 
of  which  they  were  composed,  the  latter  being  considered 
single  columns. 

All  the  white-pine  composite  columns  were  tested  wit! 


Plate  F. 


flat  ends  and  were  built  up  with  the  greatest  widths  of 
individual  sticks  adjacent  to  each  other. 

The  results  in  Tables  III  and  IV  are  shown  graphically 
in  Plate  F.  One  ordinate  gives  the  values  of  /  -r-  d,  and  the 
other  the  ultimate  resistance  in  pounds  per  sq.  in. 

The  full  curved  lines  running  into  horizontal  tangents  at 
the  left  represent  about  mean  lines  through  the  points 
indicating  the  actual  column  tests. 

The  broken  lines  represent  the  following  empirical  for- 
mulae ;  in  which  p  is  either  the  ultimate  resistance  or  work 
mg  stress  in  pounds  per  sq.  in. 


Art.  86.]  TIMBER  COLUMNS.  537 

For  yellow  pine     .     .     .     ^  =  5800— 
"    white  .     .     .     £  =  38oo- 

For  wooden  railway  structures  there  may  be  used: 

For  yellow  pine     .     .     .     p 
'•    white        "...     £ 

For  temporary  structures,  such  as  bridge  false  works 

carrying  no  traffic: 

For  yellow  pine     .     .     .     £  =  1500  —  i8(/-f-  d) 
"    white       "       .     .     .     p  =  looo-  i2(l  ^d) 

The  preceding  formula  are  to  be  used  only  between  the 

limits  of  -  =  20  and  -  =  60   for  yellow  pine  and  3=30  and 
d  d  a 

-  =  60  for  white  pine. 
d 

For  short  columns  below  -  =  20  and  -=30  there  are  to 

d  d 

be  used  for  yellow  and  white  pine  respectively  : 


Ultimate.         Railway  Bridges. 
Yellow  pine  .  .  .  .  p  =  4400  ........  550  ........  1  100  Ibs.  per  sq.  in. 

White      '  '    .  .  .  .p  =  2400  ........  300  ........   600    '  '     "        '  ' 

All  the  preceding  values  are  applicable  to  good  average 
lumber  for  the  engineering  purposes  indicated. 

Table  V  exhibits  a  number  of  results  of  the  tests  of 
short  timber  columns  taken  from  the  "  U.  S.  Reports  of 
Tests  of  Metals  and  Other  Materials"  for  1894,  1896,  1897, 
and  1900.  It  will  be  observed  that  the  ratios  of  length  over 
thickness,  i.e.,  minimum  dimension  of  cross-section,  are 
less  than  22,  and  with  two  exceptions  much  less.  These 
columns  do  not,  therefore,  come  within  the  range  of  appli- 
cation of  such  formulas  as  those  given  on  the  preceding 
page  for  yellow  pine  and  white  pine. 


538 


LONG  COLUMNS. 


[Ch.  X. 


TABLE  V. 

SHORT  TIMBER  COLUMNS. 


Timber. 

Dimensions,  Inches. 

/  . 

d 

I  2 
12 
I  2 
12 
I  2 
12 
I  2 
14 

15 

8 
9 
14 

22 
20 
15 
15 

Ultimate  Compressive 
Resistance, 
Lbs.  per  Sq.  In. 

1 
"o 

6 
% 

Leng'h 

Bre'dth 

9.8 
9.8 
9.8 
9.8 
9-8 
9.8 
9.6 

8"5 
9-5 
9-5 

4  to  6 
M 
8&  14 

I  2 
10 

Thick 

ness. 

Max. 

Mean. 

Min. 

Long-leaf  pine.  .  . 
Short-leaf  pine  .  . 

Spruce.  .  s  
Long-leaf  pine  *  ... 
Cypress  
White  pine  

1  20 

120 
120 
120 
120 
120 
120 
131 
I  2O 
58 
71 
48 
60 
60 
60 
121 

9.8 
9.8 
9.8 
9.8 
9.8 
9.8 
9.6 
9-5 
8 
7-9 
7-9 
3-5 

2.8 

3 
4-  1 
8 

4,976 
3,800 
4,200 
3,925 
3,4oo 
4,000 
3,i74 
7,354 
3,457 

4,574 
3,558 
3,957 
3,48i 
3,ooo 
3,568 
2,589 
6,093 
3,3o8 
3,652 
2,917 
5,160 

6,211 

6,725 

6,220 
3,697 
4,214 
4,372 
4,042 

4,200 

3,369 
3,714 
3,037 
2,600 
3,135 
1,900 
4,960 
3,113 

5 
5 

i 
i 

Butt      sticks. 
Top 
Middle 
Butt 
Top 
Middle 

Old  posts. 

J  Probably 
>-  170  years 

Fold. 

6,247 
7,882 

2,917 

5,568 

4,197 

Red  oak  
Douglas  fir 

White  oak  

"  '.'.'.'.'.'.'. 

74 
69 

7-5 
10 

I9 

10 

9 

*  Well  seasoned  and  dry;  12  years  old.  Had  been  in  a  fire  and  corners  were  partially 
charred. 

All  posts  represented  in  this  table  contained  probably  1 5  to  1 8  per  cent,  of  moisture,  or 
perhaps  more. 

The  long-leaf  and  short-leaf  pine  tests  show  that  columns 
taken  from  the  butts  of  trees  are  stronger  than  those 
taken  either  from  the  middle  or  the  tops,  the  top  sticks, 
as  a  rule,  having  the  least  ultimate  resistance  per  square 
inch  of  all.  The  white-pine  and  red-oak  sticks  yield  interest- 
ing results  on  account  of  their  age,  as  they  were  taken  from 
some  wooden  trusses  of  the  Old  South  Church,  Boston, 
Mass.,  a  building  constructed  in  1729.  The  timber  was  so 
housed  as  to  be  completely  protected  and  kept  very  dry. 
The  results  show  no  loss  of  resistance  as  compared  with 
tests  of  the  same  kind  of  timber  at  the  present  time. 

The  effect  of  immersion  in  water  on  the  resistance  of 
timber  is  illustrated  by  tests  made  at  the  Watertown 
Arsenal.  A  post  similar  to  one  of  the  old  long-leaf  pine 
columns,  12  of  which  were  tested  in  a  seasoned  condition 


Art.  86.]  TIMBER  COLUMNS.  539 

giving  the  average  shown  in  the  Table  of  6093  pounds  per 
square  inch,  was  submerged  in  water  for  a  period  of  130 
days  and  then  tested  with  the  result  of  failing  at  3800 
pounds  per  square  inch. 

The  values  given  in  Table  V  correspond  closely  to  the 
results  shown  for  yellow  pine  and  white  pine  on  pages 
534  and  535,  so  far  as  they  may  properly  be  compared. 


CHAPTER   XI. 
SHEARING  AND  TORSION. 

Art.  87. — Modulus  of  Elasticity. 

IT  has  already  been  shown  in  some  of  the  Articles  of 
the  first  part  of  this  book  that  the  stresses  of  shearing  and 
torsion  are  identical,  both  being  shears;  hence  the  modulus 
of  elasticity  is  the  same  for  both. 

As  it  is  much  more  convenient  to  make  accurate  deter- 
minations of  the  modulus  of  elasticity  in  torsion  than  in 
direct  shearing,  the  former  method  has  been  employed  in 
practically  all  cases.  A  number  of  such  moduli  for  four 
varieties  of  steel  are  given  in  Art.  38.  Those  values  show 
that  the  modulus  changes  but  little  for  the  different  varieties 
of  steel  indicated. 

The  aggregate  of  torsion  tests  so  far  as  they  have  been 
made  indicate  that  the  two  moduli  of  elasticity,  G  for  shear 
and  E  for  direct  stresses  of  tension  and  compression,  have 
the  approximate  relation : 

G  =  (.4  to  . 

Prof.  Bauschinger  published  in  "  Der  Civilingenieur," 
Heft  2,  i .88 1,  the  results  of  some  of  his  tests  of  cast-iron 
cylinders  or  prisms  which  are  still  valuable  on  account  of 
the  accuracy  with  which  he  made  his  determinations. 
The  prisms  were  about  40  inches  long,  and  were  subjected 
to  torsion,  while  the  twisting  of  two  sections  about  20  inches 

540 


Art.  87.] 


MODULUS  OF  ELASTICITY. 


541 


apart,  in  reference  to  each  other,  was  carefully  observed. 
The  results  for  four  different  cross-sections  will  be  given — 
i.e.,  circular,  square,  elliptical  (the  greater  axis  was  twice  the 
less),  and  rectangular  (the  greater  side  was  twice  the  less). 
In  each  case  the  area  of  cross-section  was  about  7.75  square 
inches.  The  angle  a  is  the  angle  of  torsion — i.e.,  the 
angle  twisted  or  turned  through  by  a  longitudinal  fibre 
whose  length  is  unity  and  which  is  at  unit's  distance  from 
the  axis  of  the  bar. 


Section. 
Circular.  .. 


G. 


0.007  degree .7,466,000  Ibs.  per  sq.  in. 


Elliptical 

Square 

Rectangular.  . 


0.07 

0.009 

0.076 

0.008 

0.073 

o.oi 

0.08 


.6,157,000 

7,437,000 

6,228,000 

7,039,000 

5,987,000 

6,996,000 

5,716,000 


The  formula  by  which  G  is  computed,  when  the  torsional 
moment  and  angle  a  are  given,  is  the  following: 

r    M     Ip 

~^'CA»'     '     '     '     *     '     '     (l) 

in  which  M  is  the  twisting  moment,  A  the  area  of  the  cross- 
section,  Ip  the  polar  moment  of  inertia  of  that  cross-section, 
and  c  a  coefficient  which  has  the  following  value? 

47: 2  =  39.48  for  circle  and  ellipse, 
42.70  "   square, 
42.00  "  rectangle, 

as  shown  in  Appendix  I. 

Bauschinger's  experiments  show  that  the  coefficient  of 
shearing  elasticity  for  cast  iron  may  be  taken  from  6,000,000 
to  7,000,000  pounds  per  square  inch;  also  that  it  varies  for 
different  ratios  between  stress  and  strain. 

It  has  been  shown  in  Art.  6,  that  if  E  is  the  coefficient 
of  elasticity  for  direct  stress,  and  r  the  ratio  between  direct 


542  SHEARING  AND   TORSION.  [Ch.  XI. 

and  lateral  strains,  for  tension  and  compression,  that  G 
may  have  the  following  value: 

E 
^oTTZTv       ......     (2) 


Prof.  Bauschinger,  in  the  experiments  just  mentioned, 
measured  the  direct  strain  for  a  length  of  about  4  inches, 
and  the  accompanying  lateral  strain  along  the  greater  axis 
of  the  elliptical  and  rectangular  cross-sections,  and  thus 
determined  the  ratio  r  between  the  direct  and  lateral  strains 
per  unit  in  each  direction.  The  following  were  the  results : 

COMPRESSION. 
Section.  r.  G. 

Circular o.  22 6,541,000  Ibs.  per  sq.  in 

Elliptical 0.23 6,541,000    "      "        " 

Square 0.24 6,442,000    "      "        " 

Rectangular 0.24 6,499,000    "      "        " 

TENSION. 

Circular o.  23 6,570,000  Ibs.  per  sq.  in. 

Elliptical 0.21 6,811,000    "      "        " 

Square....- 0.26 6,399,000    "      "        " 

Rectangular 0.22 6,527,000    "      "        " 

•  "  '!• 

The  values  of  E  are  not  reproduced,  but  they  can  be 
calculated  indirectly  from  eq.  (2)  if  desired. 

It  is  seen  that  the  values  of  G,  as  determined  by  the 
different  methods,  agree  in  a  very  satisfactory  manner, 
and  thus  furnish  experimental  confirmation  of  the  funda- 
mental equations  of  the  mathematical  theory  of  elasticity 
in  solid  bodies. 

The  fact  that  G  is  essentially  the  same  for  all  sections  is 
also  strongly  confirmatory  of  the  theory  of  torsion  in 
particular. 

These  experiments  show  that,  for  cast  iron,  the  lateral 
strains  are  a  little  less  than  one  quarter  of  the  direct  strains. 
If  r  were  one  quarter,  then  G  =\E,  or  E  =%G. 


Art.  88.]  ULTIMATE  RESISTANCE.  543 

Art.  88. — Ultimate  Resistance. 

It  has  seemed  more  convenient  to  give  some  values  of 
ultimate  and  working  resistances  for  the  materials  iron  and 
steel  which  are  much  more  commonly  used  than  any  others 
to  resist  torsion  in  Arts.  37  and  38,  where  the  complete 
analyses  of  the  formulae  for  the  common  theory  of  torsion 
are  given.  Those  articles  should,  therefore,  be  consulted 
for  such  formulae  and  analytic  operations  as  are  involved 
in  the  design  of  shafting  to  resist  torsion.  The  experimental 
values  set  forth  in  the  following  articles  may  be  employed 
in  the  formulae  of  the  common  theory  of  torsion  for  any  de- 
sired practical  operation  in  the  design  of  torsion  members. 

Before  considering  the  ultimate  shearing  resistance  of 
special  materials  it  will  be  well  to  notice  the  two  different 
methods  in  which  a  piece  may  be  ruptured  by  shearing. 

If  the  dimensions  of  the  piece  in  which  the  shearing  force 
or  stress  acts  are  very  small,  i.e.,  if  the  piece  is  very  thin, 
the  case  is  said  to  be  that  of  "simultaneous"  shearing.  If 
the  piece  is  thick,  so  that  those  portions  near  the  jaws  of 
the  shear  begin  to  be  separated  before  those  at  some  dis- 
tance from  it,  the  case  is  said  to  be  that  of  "shearing  in 
detail."  In  the  latter  case  failure  extends  gradually,  and 
in  the  former  takes  place  simultaneously  over  the  surface 
of  separation.  Other  things  being  the  same,  the  latter 
case  (shearing  in  detail),  will  give  the  least  ultimate  shearing 
resistance  per  unit  of  the  whole  surface. 

In  reality  no  plate  used  by  the  engineer  is  so  thin  that 
the  shearing  is  absolutely  simultaneous,  though  in  many 
cases  it  may  be  essentially  so. 

Wrought  Iron. 

There  may  be  found  in  the  Articles  on  Riveted  Joints 
some  experimental  determinations  of  the  ultimate  shearing 


544  SHEARING  AND  TORSION.  [Ch.  XI. 

resistance  of  wrought  iron  which,  under  the  conditions  of 
such  joints,  may  range  from  about  34,000  to  about  43,000 
pounds  per  square  inch.  It  has  been  observed  in  the 
consideration  of  riveted  joints  that  the  ultimate  resistance 
to  shear  of  rivets  will  generally  be  less  with  thick  plates 
than  with  thin,  because  the  bending  stresses  of  tension 
and  compression  will  generally  be  greater  for  thick  plates 
than  for  those  that  are  thinner.  If  the  riveted  joint  is  so 
designed  that  the  bending  stresses  are  not  greater  for  thick 
plates  than  for  thin  ones,  the  effects  of  bending  will  neces- 
sarily disappear. 

Such  tests  as  have  been  made  on  direct  shearing  resist- 
ance show  that  generally  it  may  safely  be  taken  at  35,000 
to  40,000  pounds  per  square  inch,  or  if  5  is  the  ultimate 
shear  per  square  inch  and  T  the  ultimate  tensile  resistance 
of  wrought  iron  per  square  inch,  there  may  be  taken  ap- 
proximately 

S  =  .ST. 

Cast  Iron. 

There  are  few  tests  available  for  the  determination  of 
the  ultimate  shearing  resistance  of  cast  iron.  For  the  ordi- 
nary grades,  such  as  cast-iron  water  pipes  and  similar  soft 
gray -iron  castings,  the  ultimate  shearing  resistance  has 
sometimes  been  taken  equal  to  the  ultimate  tensile  resist- 
ance, i.e.,  15,000  to  18,000  pounds  per  square  inch,  but 
this  is  probably  too  high  except  for  the  special  stronger 
grades  of  material. 

For  general  purposes  it  is  probably  safe  to  take  the  ulti- 
mate shearing  resistance  of  cast  iron  about  three-quarters 
of  its  ultimate  tensile  resistance.  It  should  only  be  used 
for  shearing,  however,  at  a  low  working  stress,  depending 
obviously  on  the  purpose  for  which  its  use  is  contemplated. 


Art.  88.]  ULTIMATE  RESISTANCE.  545 

Steel. 

The  results  of  Prof.  Ricketts'  shearing  tests  on  both  open- 
hearth  and  Bessemer  steel  rounds  with  different  grades  of 
carbon  are  given  in  Table  I  of  Art.  43.  The  elastic  limit 
is  the  point  at  which  the  metal  first  fails  to  sustain  the  scale 
beam.  The  double-shear  resistance  in  one  case  exceeds  the 
single  by  over  six  per  cent.  According  to  these  tests,  the 
ultimate  shearing  resistance  of  mild  steel  may  be  taken 
at  three  quarters  of  its  ultimate  tensile  resistance.  Each 
shear  result  is  a  mean  of  three  tests. 

The  rivet  steel  was  low,  containing  but  .09  per  cent,  of  car- 
bon. While  the  specimens  of  Bessemer  steel  were  a  little 
higher  in  carbon,  ranging  from  .  1 1  to  .  1 7  per  cent.,  except  the 
last  six,  they  were  also  of  low  or  medium  steel.  It  should 
be  carefully  noted  that  the  results  in  that  table  show  that 
the  ultimate  shearing  resistances  for  the  low  or  medium 
steels  running  from  44,600  pounds  per  square  inch  up  to 
53,260  pounds  per  square  inch  are  closely  three  fourths 
the  corresponding  ultimate  tensile  resistances.  On  the 
other  hand,  the  six  specimens  of  high  steel  give  ultimate 
shearing  resistances  but  little  over  two  thirds  of  the  corre- 
sponding ultimate  tensile  resistances.  This  is  a  feature  of 
the  relation  between  the  ultimate  shearing  and  ultimate 
tensile  resistances  of  different  grades  of  steel  which  is 
commonly  exhibited  in  tests.  The  high  steel  appears  to 
yield  an  ultimate  shearing  resistance  of  sensibly  less  per- 
centage of  the  tensile  ultimate  than  low  steel. 

In  the  .Arts.  74  and  76  on  riveted  joints  there  will  be 
found  a  number  of  values  of  ultimate  resistance  for  steel 
rivets  in  shear.  They  constitute  important  determinations 
of  the  ultimate  shearing  resistance  of  steel  rivets  under  con- 
ditions in  which  they  are  frequently  used. 


546 


SHEARING  AND  TORSION. 


[Ch.  XI. 


Copper,  Tin,  Zinc,  and  Their  Alloys. 

The  following  values  of  the  ultimate  resistance  to  torsive 
shear  Tm,  were  determined  by  Prof.  R.  H.  Thurston  in  his 
early  experimental  work  on  the  bronzes.  Although  these 
determinations  were  made  on  test  specimens  only  .625  inch 
in  diameter  and  with  a  torsion  length  of  i  inch,  they  con- 
stitute practically  the  only  fairly  complete  shear  and  torsion 
data  on  the  copper- tin  and  copper-zinc  alloys. 

TABLE  I. 


Composition. 

Ultimate  Torsive 
Shear,  Tm. 

Elastic  Limit, 
Per  Cent  of  Tm- 

Ultimate  Torsion 
Angle. 

Cu. 

Sn. 

Pounds. 

Degrees. 

IOO 

oo 

35,910 

35 

153-0 

100 

00 

28,430 

40 

52  to  1.54 

00 

IOO 

3,196 

45 

557-0 

00 

IOO 

3,297 

33 

691.0 

90 

10 

43,943 

4i 

114.5 

80 

20 

47,67i 

62 

16.3 

70 

30 

4,407 

IOO 

i-5 

62 

38 

1,770 

IOO 

I  .0 

52  . 

48 

686 

IOO 

I.O 

39 

61 

5,88! 

IOO 

i-7 

29 

7i 

5,257 

IOO 

2-34 

10 

90 

5,76i 

63 

131.8 

90 

10 

25,027 

49 

57-2 

90 

IO 

31,851 

57 

72.6 

Tm  is  in  pounds  per  square  inch. 

Table  I  relates  to  alloys  of  copper  and  tin,  and  Table  II 
to  alloys  of  copper  and  zinc. 

None  but  specimens  with  circular  sections  were  tested. 

An  examination  of  the  results  given  in  Tables  I  and  II 
show  that  the  resisting  capacities  of  each  series  of  alloys 
vary  greatly  with  the  varying  elements  constituting  the 
alloy.  Indeed,  the  shearing  resistances  of  these  alloys  in 
torsion  are  seen  to  vary  as  widely  as  their  tensile  resistances. 


Art.  88.] 


ULTIMATE   RESISTANCE. 
TABLE  II. 


547 


Percentage  of 

Ultimate  Torsive 

Elastic  Limit, 

Ultimate  Torsion 

Shear,  Tm 

Per  Cent  of  Tm. 

Angle. 

Copper. 

Zinc, 

Pounds. 

Degrees. 

90.56 

9.42 

35,ioo 

17.2 

458.0 

81.90 

17.99 

4i,575 

27.5 

345-0 

71  .20 

28.54 

41,000 

24.0 

269.0 

60.94 

38.65 

48,520 

29.4 

202.0 

55-15 

44-44 

52,320 

32.7       . 

109.0 

49.66 

50.14 

43,154 

36.0 

38.0 

4I-30 

58.12 

4,588 

IOO.O 

1.8 

32.94 

66.23 

7,241 

IOO.O 

I  .2 

20.81 

77.63 

16,374 

IOO.O 

0.8 

10.30 

88.88 

22,500 

85.6 

7-1 

o.oo 

IOO.OO 

9,186 

38.1 

I4I.5 

Although  the  values  of  Tm  are  the  ultimate  intensities 
of  torsive  shear,  they  may  be  accepted  as  ultimate  resist- 
ances for  direct  shear  for  the  same  alloys. 

Timber. 

The  shearing  resistance  of  timber  is  least  along  planes 
parallel  to  the  fibres  and  greatest  when  the  shearing  force 
acts  in  planes  at  right  angles  to  the  fibres.  Again,  the 
shearing  resistance  parallel  to  the  fibres  is  somewhat  differ- 
ent in  short  blocks  from  that  found  in  full-size  beams  sub- 
jected to  flexure.  In  the  latter  case  it  has  been  shown  in 
Art.  15  that  the  greatest  intensity  of  shearing  stre  s 
parallel  to  the  fibres  will  take  place  in  the  neutral  surface. 
It  has  been  found  that  for  relatively  short  spans  timber 
beams  in  flexure  will  fail  by  shear  along  the  neutral  surface. 
Hence  the  ultimate  resistance  to  shear  along  that  surface 
has  much  practical  value  and  it  has  been  determined  in 
tests  of  many  full-size  beams.  Among  the  latter  those  made 
by  Prof.  Arthur  N.  Talbot  at  the  University  of  Illinois  and 


548 


SHEARING  AND  TORSION. 


[Ch.  XI. 


described  in  the  University  of  Illinois  Bulletin  No.  15, 
December,  1909,  are  of  unusual  value.  The  full-size  beams 
were  13.5  feet  to  about  14.5  feet  span  and  with  cross-sections 
of  7  inches  by  12  inches,  7  inches  by  14  inches  and  7  inches 
by  1 6  inches.  Other  smaller  beams  were,  however,  used. 
The  beams  were  of  sound  merchantable  lumber  and  of  about 
the  quality  used  in  good  engineering  work.  The  following 
table  gives  the  results  of  these  tests,  showing  the  number 
of  pieces  tested  to  failure  with  the  highest,  average  and  lowest 
ultimate  shear  per  square  inch  along  the  fibres  in  or  near 
the  neutral  surface. 

TABLE  III. 

ULTIMATE    RESISTANCES    ARE    GIVEN    IN    POUNDS    PER 
SQUARE   INCH 


Timber. 

No.  of 

Pieces. 

Ultimate  Shearing  Stress. 

Highest. 

Average. 

Lowest. 

Untreated  longleaf  pine            

25 

4 
6 

8 

10 
10 

ii 

497 
505 
410 

391 

388 

383 
401 

370 
364 
302 

273 
3H 
298 

323 

1  88 

293 
224 
224 
253 

221 

275 

Untreated  shortleaf  pine  
Creosoted  shortleaf  pine  

Creosoted  loblolly  pine  

Untreated  loblolly  pine 

Old  Douglas  fir 

New  Douglas  fir 

The  values  given  in  Table  III  are  somewhat  smaller 
than  those  which  Prof.  Talbot  found  for  short  blocks.  The 
ultimate  shearing  stress  along  the  fibres  of  the  neutral 
axis  of  the  full-size  beams  ranged  from  75  per  cent,  up  to 
1 01  per  cent,  of  the  corresponding  results  for  short  blocks 
of  the  same  kind  of  timber.  The  new  Douglas  fir  gave  the 
highest  of  these  percentages  and  untreated  longleaf  pine 
together  with  creosoted  loblolly  pine  gave  the  smallest. 

The  American  Railway  Engineering  and  Maintenance  of 
Way  Association  has  recommended  ultimate  and  working 


Art.  88.] 


-ULTIMATE  RESISTANCE. 


549 


values  for  shear  along  the  grain  and  in  the  neutral  surface 
of  beams  as  given  in  Table  IV  of  Art.  90. 

Natural  Stones. 

The  ultimate  shearing  resistance  of  stones  has  not  as 
great  practical  value  as  the  ultimate  compressive  or  the 
ultimate  bending  resistance,  yet  there  are  occasional 
structural  conditions  under  which  it  is  necessary  to  ascer- 
tain what  shearing  capacity  may  be  relied  upon.  Valuable 
data  for  this  purpose  are  shown  in  Table  IV  taken  from  the 
"  U.  S.  Report  of  Tests  of  Metals  and  Other  Materials"  for 
1894  and  1899.  The  sheared  surfaces  were  about  6  inches 
by  4  inches  in  area.  Generally  one  such  surface  was 
sheared,  but  occasionally  two. 

TABLE  IV. 
SHEARING  RESISTANCE  OF  NATURAL  STONES. 


Stone. 

Ultimate  Shearing  Resistance,  Lbs. 
per  Square  Inch. 

Maximum. 

Mean. 

Minimum. 

Brandford  granite  Conn 

1,925 
2,872 
2,231 

2,047 

1,834 
2,554 
2,219 
,825 
,549 
,369 
,237 
,411 
,242 
,332 
,490 
,705 
,831 
'  ,204 
,150 
,243 
,352 
2,127 

1,742 
2,236 
2,197 

1,052 

1,163 
1,426 
1,389 

Milford  Granite  Mass         •                    .  . 

Troy  granite  N  H                               . 

Milford  pink  granite  Mass                  .  . 

Pigeon  Hill  granite  Mass                  .... 

Creole  marble  Ga                        

Cherokee  marble  Ga              

Etowah  marble  Ga                  

1,501 

i,554 
2,016 

Kennesaw  marble  Ga              

Marble  Hill  marble  Ga              

Tuckahoe  marble  NY             

Mount  Vernon  limestone  Ky       

Maynard  <=andctone  Mass         

1,287 
1,308 
1,383 

2,518 

1,  1  2O 
992 
I,IO2 

i,735 

Kibbe  sandstone  Mass              

\Vorcester  sandstone  Mass       

Chuckanut  candstone  ^^a^h             

Yammerthal  limestone,  Buffalo  

SHEARING  AND  TORSION. 


[Ch.  XI. 


All  the  results  except  the  last  are  taken  from  the  Report 
for  1894.  Where  but  one  value  appears  in  the  table  one 
test  only  was  made.  In  the  other  cases  two  tests  were 
made  and  the  mean  values  are  means  of  the  two  shown  in 
the  columns  containing  the  greatest  and  least.  It  will  be 
observed  that  the  ultimate  shearing  resistance  is  scarcely 
more  than  ten  per  cent,  of  the  ultimate  compressive  re- 
sistance of  the  various  stones  tested. 

The  greatest  permissible  working  stresses  for  natural 
stones  in  shear,  in  design  work,  will  necessarily  depend 
on  the  duty  to  be  performed.  In  view  of  the  variable  char- 
acter of  even  the  best  of  natural  stones  as  delivered  ready 
for  use,  one  eighth  to  one  tenth  of  the  ultimate  is  as  much  as 
should  be  taken  in  ordinary  cases,  and  materially  less  than  that 
under  some  conditions. 

Bricks. 

The  shearing  resistance  of  bricks,  like  that  of  natural 
stones,  is  seldom  employed,  but  it  is  sometimes  needed. 
The  ultimate  resistances  of  bricks  in  shearing  shown  in 
Table  V  are  taken  from  the  "  U.  S.  Report  of  Tests  of 
Metals  and  Other  Materials  "  for  1894. 

TABLE   V. 

BRICKS  IN  SHEARING. 


Kind  of  Brick. 

Ultimate 
Shearing 
Resist- 
ance, Lbs. 
per  Square 
Inch. 

Number 
of 
Sheared 
Sur- 
faces. 

Hydraulic  Press  Brick  Co. 

Northern  Hydraulic  Press 

Eastern 
«                  «             « 

ft                 ti             « 
Philadelphia  and  Boston  ] 

,  St.  Louis,  No.  6  

I,OII 
642 

1,047 
784 

7H 
1,167 

1,097 
988 

433 
639 

I 
2 

2 
2 

I 

"            "    511  

"          brown  

Chicago,  red  

Brick  Co.,  Minneapolis,  dark  red.  . 
"        "    Philadelphia,  210  .. 

220  

390  

?ace  Brick  Co..  Boston,  gray.  . 

Art.  88.]  ULTIMATE  RESISTANCE.  55! 

In  these  shearing  tests  the  sheared  surfaces  were  each 
about  2.25  by  4  inches  in  dimensions. 

The  ultimate  shearing  resistances  in  Table  V  range 
scarcely  10  to  20  per  cent,  of  the  ultimate  compressive  resist- 
ances of  the  same  materials  shown  in  Art.  68. 

Working  shearing  stresses  for  design  operations  should 
not  be  taken  more  than  one  eighth  to  one  tenth  of  the 
ultimate  values  found  in  Table  V, 


CHAPTER  XII. 

BENDING  OR  FLEXURE. 

Art.  89. — Modulus  of  Elasticity. 

THE  modulus  of  elasticity  as  determined  by  experiments 
in  flexure  can  scarcely  be  considered  other  than  a  con- 
ventional quantity.  If  the  span  of  a  beam  were  very  long 
compared  with  the  depth  of  the  beam  and  if  the  moduli 
of  elasticity  for  tension  and  compression  were  equal  to  each 
other,  and  if  all  the  hypotheses  involved  in  the  common 
theory  of  flexure  were  true,  then  the  modulus  of  elasticity 
for  flexure  would  be  a  real  quantity  and  essentially  the  same, 
at  least,  as  that  for  either  tension  or  compression. 

These  conditions,  however,  do  not  exist  in  bent  beams 
and  the  quantity  ordinarily  called  the  modulus  of  elas- 
ticity in  flexure  possesses  value  chiefly  as  an  empirical 
factor  which  enables  deflection,  independently  of  shear,  to 
be  estimated  with  sufficient  accuracy  for  all  usual  purposes. 

The  formulae  to  be  employed  in  the  determination  of 
the  modulus  of  elasticity  for  flexure  have  already  been 
established  in  connection  with  the  common  theory  of  flexure 
and  their  use  will  be  shown  in  succeeding  articles. 

Art.  90. — Formulae  for  Rupture. 

The  formulae  of  the  common  theory  of  flexure,  available 
for  practical  use,  are  true  only  within  the  limits  of  elas- 
ticity. In  the  testing  of  beams  to  failure  they  are  employed 
precisely  as  if  the  elastic  properties  of  the  material  were 
maintained  up  to  the  degree  of  loading  which  causes  failure. 

552 


Art.  90.]  FORMULA  FOR  RUPTURE.  553 

While  this,  strictly  speaking,  is  irrational,  it  is  the  only 
satisfactory  procedure  available.  By  placing  the  analytic 
expression  for  the  moment  of  the  internal  stresses  in  the 
normal  section  of  a  bent  beam  equal  to  the  moment  of  the 
external  loading  causing  failure,  the  resulting  equation  may 
be  solved  so  as  to  give  the  apparent  ultimate  intensity  of 
stress  k  in  the  extreme  fibres  of  the  beam.  The  so- 
called  intensity  of  fibre  stress  found  in  this  manner  is  an 
empirical  quantity  which  may  be  introduced  into  the  for- 
mulae of  the  common  theory  of  flexure  and  so  make  them 
applicable  to  the  operations  of  engineering  practice  in  con- 
nection with  loaded  beams  of  any  shape  of  cross-section. 

If  k  and  k1  are  the  greatest  intensities  of  stress  in  the 
section  of  rupture  and  at  the  instant  of  rupture;  y  the 
variable  normal  distance  of  any  fibre  from  the  neutral  sur- 
face; y\  and  y'  the  greatest  values  of  y\  b  the  variable 
width  of  the  section  (normal  to  y)\  and  M  the  resisting 
moment  at  the  instant  of  rupture;  then  the  general  for- 
mula for  rupture  by  bending,  as  given  by  eq.  (i)  of  Art. 
26,  is 


M=~       y*bdy+—.\    y2bdy.  .  ,,:-    •     .     (i) 

yijQ  '         y  J-y' 

This  equation  is  in  reality,  based  on  the  supposition  that 
the  moduli  of  elasticity  for  tension  and  compression  are  not 
equal.  It  is  rare,  however,  that  such  a  supposition  is  made. 
It  is  practically  the  invariable  rule  to  assume  the  moduli 
of  elasticity  for  tension  and  compression  to  have  equal 
values  and  such  an  assumption  is  fortunately  sufficiently 
accurate  for  all  ordinary  purposes. 

If  the  tensile  and  compressive  moduli  of  elasticity  are 

k     kf 
the  same  —  =—.  and  eq.  (i)  becomes 

yi   y 

M=^.  .     .     (2) 


554  BENDING  OR  FLEXURE.  [Ch.  XII. 

This  is  the  usual  equation  of  flexure  employed  so  fre- 
quently in  connection  with  the  design  of  bent  beams  or  the 
investigation  of  their  carrying  capacity,  I  being  the  moment 
of  inertia  of  the  normal  section  of  the  beam  d\  the  distance 
of  the  most  remote  fibre  from  the  neutral  axis  of  the  section 
and  M  the  moment  of  the  external  forces  or  loading  about 
the  neutral  axis  of  the  section  in  question.  In  the  prac- 
tical use  of  this  formula  it  is  only  necessary  to  introduce  the 
proper  values  of  I  and  d\  for  the  shape  of  a  section  involved. 

Art.  91.  —  Beams  with  Rectangular  and  Circular  Sections. 

These  are  the  simplest  forms  of  sections  for  bent  beams 
employed  in  engineering  work.  Timber  beams  are  with  few 
exceptions  of  rectangular  section  and  so  are  many  rein- 
forced concrete  beams,  although  in  such  a  case  the  section 
is  composite,  i.e.,  composed  of  two  materials,  and  it  will 
receive  separate  treatment  in  a  later  article.  The  solid 
circular  section  belongs  to  pins  in  pin-connected  truss 
bridges  whose  design  always  involves  their  consideration 
as  a  loaded  beam  of  very  short  span. 

The  following  are  the  values  of  I  and  d^  for  rectangular 
and  circular  sections,  h  being  the  side  of  the  rectangle  normal 
and  b  that  parallel  to  the  neutral  axis,  while  r  is  the  radius 
of  the  circular  section  and  A  the  area  in  each  case  : 


12  12 


Rectangular: 

h 


_^ **•*  , 

Circular: 


Art.  91.]  BEAMS  WITH  RECTANGULAR  AND  CIRCULAR  SECTIONS.  555 

If  the  beams  are  supported  at  each  end  and  loaded  by  a 
weight  W  at  the  centre  of  the  span  (or  distance  between 
supports)  ,  which  may  be  represented  by  /,  then  the  moment 
at  the  centre  of  the  beam  becomes 

Wl  ,  x 

£Px=M=  —  .    ......     (2) 

4 

There  will  then  result  from  eq.  (2),  Art.  89: 
For  rectangular  sections  : 


For  circular  sections  : 


The  quantity  k  is  called  the  modulus  of  rupture  for 
bending,  and  if  experiments  have  been  made,  so  that  W 
is  known,  eq.  (3)  gives 


and  eq.  (4) 

,      Wl     Wl 


If  the  rectangular  section  is  square,  bh2  =b3  =h3. 

Steel 

If  the  beam  is  simply  supported  at  each  end  and  carries 
a  load  W  at  the  centre,  while  E  is  the  coefficient  of  elasticity 
and  w  the  deflection  at  the  centre,  eq.  (28)  of  Art.  28  gives 


556  BENDING  OR  FLEXURE.  [Ch.  XII. 

If,  in  any  given  experiment,  w  is  measured,  E  may  then 
be  found  by  the  following  form  of  eq.  (7) : 


Wls 

...     (8) 


If  the  section  is  rectangular 


WP 
E=  -  ry-o  .......      (o) 

3 


These  equations  enable  the  coefficient  of  elasticity  E  to 
be  computed  readily  from  experimental  observations.  It 
is  only  necessary  to  measure  accurately  the  deflection  w 
produced  by  the  load  or  weight  W  and  then  introduce  all  the 
known  quantities  in  eq.  (8)  or  eq.  (9). 

A  bar  of  wrought  iron  3  inches  deep  and  i  inch  wide 
was  placed  on  supports  48  inches  apart  and  loaded  with  a 
weight  of  400  pounds  at  mid-span.  The  measured  de- 
flection was  .0138  inch.  Hence 

400X48X48X48 

E  =  —  —  -  =  29,730,000. 

4XiX3X3X3X.oi38 

Other  applications  may  be  made  in  precisely  the 
same  way. 

High  Extreme  Fibre  Stress  in  Short  Solid  Beams. 

During  the  period  when  wrought  iron  was  used  for 
structural  purposes,  especially  for  wrought-iron  pins  with 
diameters  up  to  9  or  10  inches,  it  was  observed  that  if  the 
ultimate  extreme  fibre  intensity  k  was  computed  by  eq.  (5) 
or  (6)  with  data  obtained  by  actual  test,  the  result  would 
be  excessively  high,  i.e.,  far  beyond  the  ultimate  resistance 
to  tension.  These  pins,  however,  on  which  are  packed  the 


Art.  91.]  BEAMS  WITH  RECTANGULAR  AND  CIRCULAR  SECTIONS.    557 

lower  chord  eye-bars  of  an  ordinary  truss  bridge,  have  very 
short  spans,  indeed  the  span  is  usually  much  less  than  the 
diameter  of  the  pin  and  sometimes  less  than  one  quarter 
of  the  diameter  of  the  pin.  It  should  be  remembered  in 
this  connection  that  the  common  theory  of  flexure  is  im- 
plicitly if  not  explicitly  based  upon  the  condition  that  the 
length  of  span  of  the  bent  beam  must  be  long  compared  with 
the  depth  of  the  beam.  In  fact  the  span  should  be  many 
times  that  depth,  and  the  longer  it  is  the  more  nearly 
correct  becomes  the  common  theory  of  flexure.  These  ob- 
servations are  equally  true  whether  the  cross-section  of  the 
beam  is  circular  or  rectangular  or  has  any  other  shape. 
The  following  Table  shows  the  results  of  tests  of  a  series 
of  short  wrought-iron  beams  of  circular  section  made  by  the 
author  when  wrought-iron  pins  were  used  in  bridge  con- 
struction, but  which  illustrate  markedly  the  intensities  of 
extreme  fibre  stress  found  with  short  spans.  It  will  be 
observed  that  the  spans  were  8  inches  and  12  inches  only 


CIRCULAR  BEAMS  OF  "BURDEN'S  BEST"  WROUGHT  IRON. 


Kind. 

Diameter. 

Span. 

W, 

K. 

Elastic. 

Ultimate. 

Elastic. 

Ultimate. 

Turned  

Ins. 
•25 
•25 
•25 
•25 
.OO 
.OO 
.OO 
.00 
.00 
.00 

0.75 
0.75 
0.75 
0.75 

Ins. 
12 

8  • 

12 

8 

12 

8    - 
13 

8 

12 

8 

12 

8 

12 

8 

Lbs. 
3,000 
4,400 

Lbs. 
6,OOO 
10,500 

Lbs. 
46,950 
45,900 
54,760 
52,150 
55,ooo 
57,000 
55,ooo 

Lbs. 
93,900 
109,500 
93,870 
114,700 
91,700 
101,900 
91,600 
107,000 
91,680 
97,800 
74,050 
85,310 
74,050 
85,310 

Turned  

Turned  

Turned 

• 

Turned 



Turned 

Rough          .    .    . 

1,700  ' 
2,800 
700 

1,200 
700 
1,300 

3,000 
4,800 
1,100 
1,900 

I,  TOO 
1,900 

5i,95o 
57,000 
47,100 
53,88o 
47,100 
58,370 

Rough          

Turned       

Turned   

Turned  

Turned  

558  BENDING  OR  FLEXURE.  [Ch.  XII. 

while  the  diameters  of  the  circular  beam  sections  varied 
from  1.25  inches  down  to  .75  inch. 

W  is  the  centre  load  and  the  extreme  fibre  intensity  k 
is  computed  by  eq.  (6).  The  ultimate  intensity  k  was 
assumed  to  be  reached  when  the  deflection  at  the  centre  of 
span  amounted  to  about  the  diameter  of  the  circular  section 
of  the  beam.  This  particular  feature  of  the  tests  is  a  matter 
of  judgment,  but  k  would  differ  little  whether  it  be  taken 
at  a  centre  deflection  equal  to  the  diameter  of  the  circular 
section  or  one  half  that  diameter  or  even  less. 

It  will  be  noticed  that  the  ultimate  values  of  k  are  all 
much  larger  for  the  8-inch  span  than  for  the  1 2-inch,  and 
that  all  the  ultimate  values  increase  materially  with  the 
depth  of  the  beam,  rising  to  107,000  to  114,700  pounds 
per  square  inch  for  diameters  (i.e.,  depths  of  beams)  of  i 
inch  and  i  J  inch.  It  will  also  be  observed  that  the  elastic 
limits  are  greatly  increased.  The  ultimate  tensile  resist- 
ance of  the  iron  used  in  these  tests  was  about  55,000 
pounds  per  square  inch  and  the  elastic  limit  a  little  more 
than  half  that  value. 

Steel. 

Investigation  by  actual  test  has  shown  that  short  steel 
beams  with  circular  or  rectangular  section  will  exhibit  the 
same  elevation  of  ultimate  intensity  of  fibre  stress  k  as 
found  for  wrought  iron  in  the  preceding  section.  This  is 
well  illustrated  by  the  following  tabular  statement  of  results 
of  tests  of  Bessemer  steel  beams  with  circular  cross-section, 
also  made  by  the  author  in  the  early  days  of  the  use  of  steel 
for  bridge  building. 

The  Table  is  self-explanatory  in  view  of  the  explanations- 
made  for  short  wrought-iron  beams  of  circular  section.  The 
ultimate  tensile  resistance  of  the  mild  Bessemer  steel  used 
in  these  tests  was  about  65,000  to  70,000  pounds  per  square 


Art.  91.]  BEAMS  WITH  RECTANGULAR  AND  CIRCULAR  SECTIONS.  559 
CIRCULAR  BESSEMER  STEEL  BEAMS,  EQ.  (6). 


Kind. 

Diameter. 

Span 

PI 

?. 

k 

Elastic. 

Ultimate. 

Elastic. 

Ultimate. 

In. 

Ins. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

I  .OO 

8 

8s  ^oo 

146,75° 
152  800 

I  .OO 

12 

2,500 

4,500 

76,400 

I  ^7    S2O 

I  .OO 

8 

3,750 

7,500 

76,400 

152  800 

0.75 

12 

1,150 

1,  800 

77,400 

122  2OO 

o.  75 

8 

1,  800 

3,300 

80,800 

148  2OO 

0.75 

12 

1,150 

1,700 

77,400 

I  14  4OO 

0.75 

8 

1,  800 

3,300 

80,800 

I48.2OO 

inch  and  the  elastic  limit  about  35,000  to  38,000  pounds 
per  square  inch.  The  ultimate  intensity  of  stress  in  the 
extreme  fibres  of  these  beams  ranged,  however,  from  114,400 
up  to  152,800  pounds  per  square  inch,  the  larger  values 
belonging  to  the  greater  depth  of  beam  and  the  smaller 
values  to  the  smaller  depth.  The  elastic  limit  is  seen  to 
be  correspondingly  high. 

These  and  the  preceding  tests  show  that  the  apparent 
ultimate  resistance  of  wrought  iron  and  structural  steel  in 
the  extreme  fibres  of  very  short  beams  with  circular  or 
rectangular  cross-section  may  be  even  more  than  twice  the 
ultimate  tensile  resistance  as  derived  from  the  testing  of 
ordinary  tensile  specimens. 

This  feature  becomes  even  more  marked  when  the 
spans  of  the  cylindrical  beams  are  still  shorter,  perhaps 
as  short  as  the  diameter  of  the  circular  section. 

In  the  design  of  pins  in  pin-connected  bridges,  this  high- 
resisting  capacity  of  wrought  iron  or  steel  in  pins  is  recog- 
nized by  making  the  working  resistance  in  the  extreme 
fibres  of  pins  considered  as  beams  as  much  as  50  per  cent, 
higher  than  in  members  subjected  to  simple  or  direct 
tension. 


560  BENDING  OR  FLEXURE.  [Ch.  XII. 

The  explanation  of  this  phenomenally  high  resistance  to 
the  tension  of  flexure  (and  also  the  compression)  is  found, 
as  already  indicated,  in  the  fact  that  the  common  theory 
of  flexure  is  not  correctly  applicable  to  such  excessively 
short  beams.  No  such  high  intensity  of  tensile  (or  com- 
pressive)  stress  actually  exists  in  the  metal  as  computed  by 
eqs.  (5)  and  (6).  When  the  span  becomes  very  short,  not 
more  than  perhaps  three  or  four  times  the  depth  of  the 
beam,  lines  of  stress  run  from  the  point  of  application  of 
the  load  at  the  centre  of  the  span  direct  to  both  supports, 
transverse  shear  being  the  vertical  components  of  the 
stresses  acting  along  these  lines.  All  such  or  similar  stress 
action  reduces  the  actual  flexure  and  makes  the  bending 
stresses  of  tension  and  compression  correspondingly  less; 
but  as  the  flexure  formulae,  eqs.  (5)  or  (6),  contain  no 
recognition  of  this  condition,  the  apparent  fibre  stresses 
computed  by  their  use  are  far  above  the  actual. 

Numerous  other  similar  short  solid  beam  tests  have 
confirmed  the  results  given  in  the  preceding  two  Tables. 

Cast'  Iron. 

Although  cast  iron  is  rarely  ever  used  to  resist  flexure 
except  in  window  and  door  lintels  or  other  similar  members 
whose  duties  are  light,  tests  of  short  cast-iron  beams  have 
shown  the  same  phenomena  of  greatly  elevated  ultimate 
resistance  as  found  for  the  more  ductile  metals.  The 
apparent  ultimate  intensity  k  in  the  extreme  fibres  of  short 
cast-iron  beams  of  circular  or  square  section  may  be  taken 
50  per  cent,  above  the  ultimate  tensile  resistance  of  the 
same  metal  under  ordinary  tensile  tests. 

Alloys  of  Aluminum. 

Table  VIII  of  Art.  59,  in  the  fifth  column  from  the  left 
side,  exhibits  values  of  the  ultimate  stress  in  the  extreme 


Art.  91.]  BEAMS  WITH  RECTANGULAR  AND  CIRCULAR  SECTIONS.    561 

fibres  of  small  beams  of  varying  proportions  of  aluminum  - 
zinc  alloys.  As  might  be  anticipated,  beams  of  either  of 
those  metals  showed  comparatively  low  resistance,  but 
with  aluminum  varying  from  80  down  to  50  per  cent,  and 
zinc  from  20  up  to  50  per  cent,  the  resistance  was  excel- 
lent, the  maximum  being  found  with  Al  75  and  zinc  25. 

Table  XI  of  Art.  59  exhibits  the  ultimate  fibre  stresses 
in  small  beams  of  the  alloys  of  aluminum  with  copper,  zinc, 
manganese  and  chromium.  The  rolled  bars  of  Al  96  and 
Cu  4  give  excellent  results;  as  does  the  cast  bar  of  A I  75.7, 
Cu  3,  zinc  20  and  Man  1.3.  The  remaining  values  of  the 
transverse  resistances  in  the  table  are  self-explanatory. 

Copper,  Tin,  Zinc,  and  their  Alloys. 

In  the  following  table  are  given  the  data  and  the  results 
of  the  experiments  of  Prof.  R.  H.  Thurston,  as  contained  in 
his  various  papers,  to  which  reference  has  already  been 
made.  The  distance  between  the  points  of  support  was 
twenty -two  inches,  while  the  bars  were  about  one  inch 
square  in  section,  and  of  cast  metal. 

The  modulus  of  rupture,  kt  is  found  by  eq.  (5),  in 
which,  however,  in  many  of  these  cases,  W  is  the  weight 
applied  at  the  centre,  added  to  half  the  weight  of  the  bar. 
When  k  is  large  and  the  specimens  small,  this  Correction 
for  the  weight  of  the  bar  is  unnecessary ;  otherwise  it  is  ad- 
visable to  introduce  it. 

The  coefficient  of  elasticity,  E,  is  found  by  eq.  (9),  in 
which  W  is  the  centre  load  added  to  five  eighths  of  the 
weight  of  the  bar. 

The  manner  in  which  both  these  corrections  arise  is  com- 
pletely shown  in  Case  2  of  Art.  28. 

E,  for  any  particular  bar,  has  a  varying  value  for  dif- 
ferent degrees  of  stress  and  strain.  Those  given  in  the  table 


562 


BENDING  OR  FLEXURE. 


[Ch.  XII. 


SQUARE  BARS. 


Percentage  of 

*, 

Lbs.  per 
Sq.  In. 

Elastic  over 
Ultimate. 

Final 
Deflection. 

E, 

Lbs.  per 
Sq.  In. 

Cu. 

Sn. 

Zn. 

Ins. 

IOO 

O.OO 

0.00 

29,850 



8.00 

9,000,000 

100 

o.oo 

0.00 

25,920 

j      0.14 

|  to  0.41 

1.38 

to  8  .  oo 

j-  10,830,600 

IOO 

o.oo 

0.00 

21,251 

0.346 

2.31 

13,986,600 

IOO 

o.oo 

0.00 

29,848 

o.  140 

Bent. 

10,203,200 

90 

IO.OO 

0.00 

49,400 

0.400 

Bent. 

14,012,135 

90 

IO.OO 

0.00 

56,375 

0.41 

3.36 



80 

20.00 

0.00 

56,715 

0.657 

0.492 

13,304,200 

70 

30.00 

0.00 

12,076 

1  .00 

0.062 

15,321,740 

61.7 

38.3 

0.00 

2,761 

1  .00 

0.032 

9,663,990 

48.0 

52.0 

0.00 

3,600 

1  .00 

0.019 

1  7,  039,  1  30 

39-2 

60.8 

0.00 

8,400 

1  .00 

0.060 

12,302,350 

28.7 

71.3 

o.oo 

8,067 

0.583 

O.  121 

9,982,832 

9-7 

90.3 

0.00 

5,305 

0.25 

Bent. 

7,665,988 

o.oo 

IOO 

o.oo 

3,740 

0.273 

Bent. 

6,734,840 

o.oo 

IOO 

0.00 

4,559 

0.267 

Bent. 

5,635,590 

80.00 

o.oo 

20.00 

21,193 



3-27 

11,000,000 

62.50 

o.oo 

37.50 

43,216 



3.13 

14,000,000 

58.22 

2.30 

39.48 

95,620 



1.99 

1  1  ,000,000 

55-00 

0.50 

44.50 

72,308 





92.32 

o.oo 

7.68 

21,784 

0.30 

Bent. 

13,842,720 

.82.93 

o.oo 

16.98 

23,197 

0.41 

Bent. 

14,425,150 

71  .  20 

o.oo 

28.54 

24,468 

0.51 

Bent. 

14,035,330 

63.44 

o.oo 

36,36 

43,216 

0.53 

Bent. 

14,101,300 

58.49 

o.oo 

41.10 

63,304 

0.48 

Bent. 

11,850,000 

54-86 

o.oo 

44.78 

47,955 

0.39 

Bent. 

10,816,050 

43-36 

o.oo 

56.22 

17,691 

I.  00 

o  .  0982 

12,918,210 

36-62 

o.oo 

62.78 

4,893 

I.  00 

0.0245 

14,121,780 

29.20 

o.oo 

70.17 

16,579 

1  .00 

o  .  0449 

14,748,170 

20.81 

o.oo 

77.63 

22,972 

1  .00 

0.1254 

14,469,650 

10.30 

o.oo 

88.88 

41,347 

0.73 

0.5456 

12,809,470 

o.oo 

o.oo 

100.00 

7,539 

0.57 

o  .  i  244 

6,984,644 

7O.  22 

8.90 

20.68 

50,541 

0.4019 

14,400,000 

56.88 

21.35 

21.39 

2,752 



0.0146 

14,800,000 

45-00 

23.75 

31.25 

6,512 



0.0150 

7,000,000* 

66.25 

23.75 

IO.OO 

8,344 



0.0162 

12,000,000* 

10.00 

50.00 

40.00 

21,525 

Bent. 

9,000,000 

58.22 

2.30 

39.48 

.  95,623 

2  .OOO 

10,600,000 

60.00 

10.00 

30.00 

24,700 

o.  1267 

14,500,000 

65.00 

20.00 

15.00 

11,932 

0.0514 

17,000,000 

70.00 

10.00 

20.00 

36,520 

0.1837 

15,000,000 

75-00 

5.00 

20.00 

55,35S 

Bent. 

13,000,000 

80.00 

10.00 

IO.OO 

67,117 

Bent. 

13,500,000 

55-00 

5-00 

44.50 

72,308 

Bent. 

11,000,000 

60.00 

2.50 

37.50 

69,508 

i  .500 

13,000,000 

72.52 

7.50 

20.00 

51,839 

Bent. 

12,000,000 

77-50 

12.50 

10.00 

61,705 

0.705 

13,500,000 

85.00 

12.50 

2-5 

62,405 



Bent. 

12,500,000 

*  These  bars  were  about  half  the  length  of  the  others. 


Art.  91.]  BEAMS  WITH  RECTANGULAR  AND  CIRCULAR  SECTIONS.  563 

may  be  considered  average  values  within  the  elastic 
limit. 

As  usual,  "elastic  over  ultimate"  is  the  ratio  of  k  at  the 
elastic  limit  over  its  ultimate  value. 

An  examination  of  the  ultimate  tensile  and  compressive 
resistances  of  these  same  alloys,  as  given  iti  preceding  pages, 
shows  that  the.  ratio  of  k  over  either  of  those  resistances  is 
very  variable.  It  is  usually  found  between  them,  but  occa- 
sionally it  exceeds  both. 

Timber  Beams. 

As  timber  beams  are  always  rectangular  in  section,  eq. 
(3)  only  will  be  needed.  Retaining  the  notation  of  that 
equation,  if  the  beam  carries  a  single  weight  W  at  the  centre 
of  trie  span  /, 

2  kAh  (    N 

W  =  -—..     ......     (10) 

If  the  total  load  W  is  uniformly  distributed  over  the 
span, 


(I!) 


As  k  is  supposed  to  be  expressed  in  pounds  per  square 
inch,  all  dimensions  in  eqs.  (10)  and  (n)  must  be  expressed 
in  inches. 

In  the  use  of  timber  beams  it  is  usually  convenient  to 
take  the  span  /  in  feet,  and  the  breadth  (b)  and  depth 
(h)  in  inches.  Placing  i2/  for  /,  therefore,  in  eqs.  (10) 
and  (n), 

T3rr     kAh  ,rl      kAh  ,     N 

W=^;     and     W  =  2^  .....     (12) 


564  BENDING  OR  FLEXURE.  [Ch.  XII. 

in  which  formulae  /  must  be  taken  in  feet  and  A  and  h  in 
inches. 

k 

If  B  be  put  for  —  t  eq.  12  becomes 
18' 

W  =  B^>     and     W'  =  2B~.    .     .     .     (13) 

Hence  when  W  and  W  have  been  determined  by  ex- 
periment, 

For  single  load  W  at  centre 

Wl  Wl      iSWl_    \Wl'  IWI 

~~        •'•  ;  ~=~Ak~      \6=4-24\       •      (I4) 


For  total  load  W  uniformly  distributed 


~  2Ah  ~2AB"  Ak     ^2Bb~6^  kb- 

it the  beam  has  a  section  one  inch  square  and  is  one  foot 

W 

long,   B  =  W '  = — .     B,   therefore,   may  be   considered   the 

unit  of  transverse  rupture ;  it  is  sometimes  called  the  coefficient 
for  centre-breaking  loads. 

If  the  depth  h  of  the  beam  is  given  and  the  breadth  is 
desired,  eq.  (14)  gives 

Wl 


Eq.  15  also  gives 

Wl       gWl 


In  general,  whatever  may  be  the  distribution  of  the  load- 
ing, if  the  bending  movement  is  M  (in  inch -pounds),  eq. 
(3)  gives 


Art.  91.)  BEAMS  WITH  RECTANGULAR  AND  CIRCULAR  SECTIONS.  565 

w       rw 


or 

b     M     6M 

- 


The  general  observations  which  have  already  been  made 
in  connection  with  the  ultimate  resistances  of  timber  in 
tension  and  compression  are  equally  applicable  to  the  flex- 
ure or  bending  of  timber  beams.  The  ultimate  resistance 
of  the  timber  as  exhibited  by  the  intensity  of  stress  in  the 
extreme  fibre  can  safely  be  taken  only  when  determined 
from  tests  of  full-size  beams  as  actually  used  in  engineering 
structures.  Such  resistances  or  moduli  when  determined 
from  small  pieces  selected  for  the  purpose  of  test  are  liable 
to  be  largely  in  error  for  the  reasons  given  in  detail  in  Art.  61. 
In  fact  Messrs.  Cline  and  Heim  state  in  Bulletin  108, 
"  Tests  of  Structural  Timbers,"  U.  S.  Department  of  Agri- 
culture, that  values  obtained  from  testing  small  thoroughly 
seasoned  selected  specimens  "  may  be  from  one  and  one 
half  to  two  times  as  high  as  stresses  developed  in  large 
timbers  and  joists,"  and  that  statement  is  rather  under  than 
over,  as  many  tests  have  shown.  Furthermore,  it  is  essen- 
tial to  know  at  least  approximately  the  degree  of  seasoning 
to  which  the  timber  has  been  subjected.  Ordinary  air 
seasoning  will  seldom  reduce  the  moisture  in  full-size  timber 
beams  to  less  than  15  per  cent,  to  20  per  cent.  Inasmuch 
as  timber  in  open  engineering  structures,  like  bridges,  will 
at  ali  times  be  exposed  to  rainfalls  often  heavy,  working 
stresses  used  in  the  design  of  such  structures  should  be 
prescribed  for  wet  or  green  condition.  If  the  structure  is 
to  be  protected  from  atmospheric  moisture,  values  belong- 
ing to  seasoned  timber  may  properly  be  employed. 

Table  II  of  Art.  61  gives  the  modulus  of  rupture  for 


566 


BENDING  OR  FLEXURE. 


[Ch.  XII. 


full-size  beams  tested  to  failure  on  a  span  of  1 5  feet  by  con- 
centrated loading  at  two  points  one  third  of  the  span  from 
each  end  (Messrs.  Cline  and  Heim,  U.  S.  Dept.  Agri- 
culture). These  results  include  failures  by  tension  and 
compression  of  fibres  as  well  as  failures  due  to  shear  along 
the  neutral  surface  of  the  beams.  Both  green  and  air- 
seasoned  timbers  were  tested  with  the  sections  given  in  the 
Article  cited. 

Table  I  gives  the  results  of  the  same  series  of  tests  under 
a  proposed  grading  by  which  all  beams  tested  were  divided 
into  Grade  I  and  Grade  II,  the  higher  resistances  being 
found  in  the  former. 

TABLE  I. 

AVERAGE   RESISTANCE   VALUES   OF   DIFFERENT   SPECIES   BY 
PROPOSED   GRADES 


Species. 

Number  of  Tests. 

Average 
Modulus  of 
Rupture  per 
Square  Inch. 

Average 
Fibre  Stress 
at  plastic 
Limit  per 
Square  Inch. 

Average  Modulus 
of  Elasticity  per 
Square  Inch. 

Total. 

Grade 

Grade 

Grade 

Grade 

Grade 

Grade 

Grade 

Grade 

I. 

II. 

I. 

II. 

I. 

II. 

I. 

II. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Longleaf  pine  .  . 
Bouglas  fir.  ... 

17 
161 

I? 
81 

'So 

6,140 
6,919 

5,564 

3,734 
4,402 

3,831 

,463,000 
,643,000 

,468,000 

Shortleaf  pine.  . 

48 

35 

13 

5,849 

•4.739 

3,318 

3,005 

,525,000 

,324,000 

Western  larch.  . 

62 

45 

17 

5,479 

3.543 

3,662 

2,432 

,365,000 

,130,000 

Loblolly  pine  .  . 

94 

45 

49 

5,898 

4.702 

3,513 

2,793 

,535,ooo 

,309,000 

Tamarack  

25 

9 

16 

5,469 

4,525 

3,151 

2,847 

,276,000 

,261,000 

West,  hemlock. 

39 

26 

13 

5,615 

4.658 

3,689 

3.172 

,481,000 

,360,000 

Redwood  

28 

21 

7 

4,932 

3.091 

4,031 

2,947 

,097,000 

877,000 

Norway  pine.  .  . 

34 

i? 

17 

4,821 

3.764 

3,082 

2,364 

,373.000 

1,204,000 

The  intensities  of  stresses  in  extreme  fibres  are  averages 
for  each  kind  of  timber  at  rupture  and  at  elastic  limit.  It 
is  to  be  understood,  however,  that  the  elastic  limit  is  approxi- 
mate only  as  it  is  not  a  well-defined  point  in  timber. 
The  moduli  of  elasticity  are  fully  as  high  as  should  be  taken, 
if,  indeed,  they  are  not  a  little  too  high  for  ordinary  pur- 
poses. 


Art.  91.]  BEAMS  WITH  RECTANGULAR  AND  CIRCULAR  SECTIONS.  567 

The  table  does  not  include  results  for  white  pine  and 
spruce,  but  the  resisting  and  elastic  qualities  of  those  two 
timbers  are  so  near  to  the  corresponding  qualities  of  Norway 
pine  that  they  may  be  assumed  to  be  the  same  under  ordi- 
nary conditions. 

Table  II  gives  a  summary  of  the  results  of  tests  of  full- 
size  beams  made  by  Prof.  Arthur  N.  Talbot  and  described 
by  him  in  Bulletin  No.  41  (1909)  of  the  University  of  Illinois. 
The  cross-sections  of  these  beams  varied  from  7  inches  by 
12  inches  to  8  inches  by  16  inches  and  the  spans  were  13.5 
feet  and  14.5  feet.  The  loads  were  applied  equally  at  two 
points,  each  one  third  of  the  span  from  each  end. 

The  series  into  which  the  program  of  results  is  divided 
were  used  as  a  matter  of  convenience  only  and  have  no  sig- 
nificance as  to  quality  of  material  or  as  to  physical  features 
of  the  results. 

It  will  be  observed  that  small  beams  and  shear  blocks 
were  also  tested  and  that  the  results  for  these  smaller  pieces 
are  on  the  whole  materially  larger  than  for  the  full-size 
beams  and  nearly  or  quite  twice  as  large  in  some  cases. 

The  extreme  fibre  stress  was  computed  by  means  of 
eq.  (5),  in  which  W  is  the  total  load  at  the  two  points  of 
application  at  failure  and  /  is  two-thirds  of  the  actual  length 
of  -span  in  the  tests,  which  makes  the  bending  moment 
M  =%Wl.  If  this  external  bending  moment  is  placed  equal 

2kl 

to  the  -r-  ,  the  intensity  of  stress  k  will  take  the  value,  as 
indicated  by  eq.  (5)  : 


In  this  equation  h  is  the  depth  of  the  beam  and  b  its 
breadth,  as  already  explained  in  connection  with  eqs.  (i) 
and  (ia).  W  is  obviously  the  load  given  by  the  reading 


TABLE  II.  % 

SUMMARY  OF  RESULTS  OF  TESTS. 
ALL  STRESSES  ARE  GIVEN  IN  LBS.  PER  SQ.  IN. 

BEN 

w               s. 

£*•      55  |h 

D/NG  OR  FLEXURE. 

[Ch.  XII 

^J    . 
co               Q 

IOOO     OivO    <N    l^-3-OOO 

*>«>+    £* 

&                                                               '        *rl 

^       c!l 

03     >, 

MMt-fOoO^-        OOiNiN    ONO 

^S-ooo'So^^ 

**                         «   ^  "* 

'g.2            ,  *2 

oa-g                 d  •£ 

Miomc»oo\        wooK^ooo 

«   *                        ^  *  « 

5.92   *o 

.5          8  <u 

^             0  g 

00   «oo   r^OO         r^oo   ^-^^0 

aroroO   ,0     '      '      '      ' 

N    CO    ^t"           MfO-MOOO 

£!  cf  o"oo  M  •  •  •  • 

n  N  ro    .    .    .    . 

§                     -^ 

••^-lOirjoO^vO           O\OOOOin(NO 

PO  ro  10  to  to    •    •    •    • 

y-*j  •  •  • 

'        £/     J| 

-IC^SS^     S2;fS£S    SJ^^^^  :  :  :  : 

"^             ^^^          ^^  -  •  •  • 

^ 

Q                       CH    +i 

W       '            jj 

O         \O^OO         vOOO\woOO            '      
M         rocooo         roiONooooO           i'l!' 

oj'ci  C 

IF 

•*nt^OMO          O^tt^-OfOO 

TfTl-OOOroOOco 

10   l^   10          0   00 

o>  o5 
<S  "3>  c 

<3  o^ 
en  J 

O    O    O    rJ-\O  00         Ot^-oOOl^-O 

«RMSSSSS 

^   (N   00          vO   00 

Number  of  stringers  tested  
Percentage  of  horizontal  shear  failures  
Percentage  of  tension  and  compression  failures.  .  . 
Average  horizontal  shearing  stress  at  failure  
Average  fibre  stress  at  elastic  limit  
Average  fibre  stress  at  failure  
Failing  by  horizontal  shear: 
Average  horizontal  shearing  stress  at  failure  .  .  . 
Highest  horizontal  shearing  stress  at  failure.  .  .  . 
Lowest  horizontal  shearing  stress  at  failure  
Highest  fibre  stress  at  failure  
Average  fibre  stress  at  failure  
Lowest  fibre  stress  at  failure  
Failing  by  tension  or  compression: 

Average  horizontal  shearing  stress  at  failure  .  .  . 
Highest  horizontal  shearing  stress  at  failure  .... 
Average  fiore  stress  at  failure  
Lowest  fibre  stress  at  failure  
Highest  fibre  stress  at  failure  
Small  beams.  Average  horizontal  shearing  stress. 
Small  beams.  Average  fibre  stress  at  elastic  limit 
Small  beams.  Average  fibre  stress  at  failure  
Shear  blocks.  Average  shearing  stress  

Art.  91.]  BEAMS  WITH  RECTANGULAR  AND  CIRCULAR  SECTIONS.  569 

of  the  scale  beam  of  the  testing  machine.  If  W\  is  one  of 
the  two  equal  loads  applied  to  the  beam  at  each  one  third 
point  of  the  span,  iW\  must  be  written  for  W. 

The  ultimate  intensity  of  shear  shown  in  Table  II,  which 
is  both  the  intensity  of  shear  in  the  neutral  surface  and  on 
a  normal  section  of  the  beam  at  the  same  point,  is  found  by 
simply  taking  one  and  one  half  the  end  reaction  divided 
by  the  cross-section  bh  of  the  beam.  .As  the  total  transverse 
shear  is  greatest  at  the  end  of  the  span,  the  greatest  inten- 
sity of  shear  on  the  neutral  surface  will  be  found  at  that 
point  at  or  near  which  failure  by  shear  will  begin  unless 
induced  elsewhere  by  a  season  crack,  wind-shake,  decay 
or  some  other  weakness  of  the  material.  Obviously  there 
is  neither  transverse  nor  longitudinal  shear  between  the 
two  points,  equally  loaded,  as  they  are  symmetrically  located 
with  reference  to  the  centre  of  the  span. 

Table  III  shows  the  moduli  of  elasticity  computed  by 
Professor  Talbot  from  the  data  secured  by  his  beam  tests. 
The  modulus  is  found  by  observing  the  centre  deflection  of 
the  beam  when  loaded  within  its  elastic  limit  and  then 
inserting  the  observed  value  of  the  deflection  and  the  cor- 
responding observed  load  in  a  formula  similar  to  eq.  (7). 
Eq.  (7)  itself  is  not  applicable  for  the  reason  that  these 


TABLE  III. 


Timber. 

Modulus  of  Elasticity  (£). 

Max. 

Mean. 

Min. 

Longleaf  pine  
Shortleaf  pine,  untreated  
Shortleaf  pine,  creosoted  
Loblolly  pine,  untreated  

2,105,000 

1,595,000 
1,478,000 
1,915,000 
1,857,000 
2,087,000 
i  ,900,000 

,620,OOO 
,591,000 
,229,000 
,386,000 
,251,000 
,780,000 

,499,000 

1,025,000 
1,585,000 
887,000 
944,000 
6lI,OOO 
1,310,000 
1,138,000 

Loblolly  pine,  creosoted  
Old  Douglas  fir 

New  Douglas  fir. 

BENDING  OR  FLEXURE. 


[Ch.  XII. 


beams  were  not  loaded  at  the  centre  of  span.  The  formula 
for  the  centre  deflection,  however,  is  readily  derived  by 
an  analysis  similar  to  that  used  in  Art.  28.  That  operation 
will  give 


i296£/' 


or     E  = 


The  preceding  experimental  values  for  timber  are  among 
the  latest  determinations  and  are  representative  of  the 
best  engineering  practice,  especially  as  they  are  based  on 
tests  of  full-size  timbers  of  as  good  quality  as  can  probably 
be  secured  in  the  open  market. 

The  American  Railway  Engineering  Association,  after 
careful  scrutiny  of  all  tests  of  timber  made  up  to  1911, 
recommended  the  values  given  in  Table  IV  for  use  in  the 

TABLE   IV. 
UNIT  STRESSES   IN   POUNDS   PER   SQUARE   INCH 


Timber. 

Bending. 

Shearing. 

Extreme  Fiber 
Stress. 

Modulus 
of 
Elasticity. 

Mean. 

Parallel  to 
the  Grain. 

Longitudinal 
Shear  in 
Beams. 

Mean 
Ult. 

Working 
Stress. 

Mean 
Ult. 

Working 
Stress. 

Mean 
Ult. 

Working 
Stress. 

Douglas  fir  

6,100 
6,500 
5,600 
4,400 
4,800 
4,200 
4,600 
5,800 
5,000 
4,800 
4,200 
5,700 

I,2OO 
1,300 
1,100 
9OO 
I,OOO 
800 
QOO 
I.IOO 
9OO 
90O 
800 
I.IOO 

,510,000 
,610,000 
,480,000 
,130,000 
,310,000 
,I9O,OOO 
,220,000 
,480,000 
800,OOO 
1,150,000 
8OO,OOO 
I,I50,OOO 

690 
720 
710 
400 
600 
590* 
670 
630 
300 
500 

840 

170 
1  80 
170 
IOO 
ISO 
130 
170 
1  60 
80 
1  2O 

210 

270 
300 
330 
1  80 
170 
250 
260 
270* 

270 

110 
120 
130 
70 
70 
IOO 
IOO 
IOO 

no 

Longleaf  pine  .... 
Shortleaf  pine.  .  .  . 
White  pine  
Spruce  
Norway  pine  
Tamarack 

Western  hemlock  . 
Redwood  
Bald  cypress  .... 
Red  cedar  
White  oak  

Unit  stresses  are  for  green  timber  and  are  to  be  used  without  increasing 
the  live  load  stresses  for  impact.  Values  noted  *  are  for  partially  air-dry 
timbers. 


Art.  91.]  BEAMS  WITH  RECTANGULAR  AND  CIRCULAR  SECTIONS.    571 

design  and  construction  of  timber  railway  structures  for 
the  modulus  of  elasticity  in  flexure,  the  ultimate  resistance 
and  working  stress  in  extreme  fibres  of  bent  beams,  and 
similar  quantities  for  ordinary  shearing  parallel  to  the  grain 
and  for  longitudinal  shearing  along  the  fibres  in  the  neutral 
surface  of  beams. 

The  intensities  of  working  stresses  given  in  this  Table 
are  for  railway  structures.  It  may  be  justifiable  to  use 
somewhat  higher  values  in  other  structures  where  the  mov- 
ing loads  are  more  steady  or  where  perhaps  it  may  be  proper 
to  consider  all  loading  as  practically  quiescent  or  dead  load. 
It  is  always  to  be  remembered,  however,  that  timber  struc- 
tures are  usually  highly  combustible  and  hence  that  it 
will  frequently  be  advisable  to  provide  some  surplus  of 
sectional  area  to  prolong  the  carrying  capacity  of  timber 
members  after  the  beginning  of  a  fire. 


Failure  of  Timber  Beams  by  Shearing  Along  the  Neutral 

Surface. 

In  the  preceding  treatment  of  timber  beams,  it  has  been 
assumed  that  when  broken  under  test  the  extreme  fibres 
will  fail,  either  in  tension  or  compression.  As  a  matter  of 
fact,  failure  of  such  beams  usually  takes  place  at  some  weak 
spot,  as  a  knot,  point  of  incipient  or  active  decay,  or  at  some 
other  point  where  abnormal  weakness  is  developed.  This 
latter  observation  holds  true  whether  the  failure  of  the  beam 
takes  place  by  tension  or  compression  in  the  extreme  fibres 
or  by  shearing  in  the  neutral  surface. 

In  Art.  15  it  was  shown  that  the  greatest  intensity  of 
either  transverse  or  longitudinal  shear  in  any  normal  sec- 
tion of  a  beam  takes  place  at  the  neutral  surface,  and  hence 
that  the  tendency  of  the  fibres  there  is  to  separate  by  longi- 


572  BENDING  OR  FLEXURE.  [Ch.  XII. 

tudinal  movement  over  each  other.  This  is  precisely  the 
kind  of  failure  which  actually  takes  place  in  some  short  tim- 
ber beams.  If  the  total  transverse  shear  at  any  normal  sec- 
tion of  the  beam  is  5,  eq.  (8)  of  Art.  15  shows  that  the 
maximum  intensity,  s,  of  shear  in  the  neutral  surface  is 


<«>> 


In  this  equation,  b  is  the  breadth  or  width  of  the  beam 
and  d  the  depth,  usually  taken  in  inches. 

If  W  is  a  single  weight  or  load  at  the  centre  of  span  of  a 
beam  simply  supported  at  each  end,  the  shear  s,  as  far  as 
that  single  load  is  concerned,  is  constant  throughout  the 
entire  length  of  the  beam  with  the  value 


If,  again,  the  beam  is  uniformly  loaded  with  the  total 
load  W't  the  intensity  of  shear  5  in  the  neutral  surface  has 
a  value  which  varies  from  zero  at  the  centre  of  span  to  the 
value  given  by  eq.  (21)  after  making  W  =  W'.  Whenever 
the  value  of  the  intensity  s  exceeds  the  ultimate  intensity 
of  shear  along  the  fibres  lying  in  the  neutral  surface,  the 
beam  will  fail  by  the  separation  of  its  two  halves  or  parts 
at  the  neutral  surface. 

The  mean  values  for  the  ultimate  resistance  to  shear 
along  the  fibres  in  the  neutral  surface  of  his  loaded  beams 
were  found  by  Prof.  Talbot  and  are  given  in  Table  II  for 
the  best  varieties  of  pine  timber  and  for  Douglas  fir,  in- 
cluding results  for  creosoted  beams  of  shortleaf  pine  and 
loblolly  pine.  The  values  for  shear  and  other  quantities 
recommended  by  the  American  Railway  Engineering  Associ- 
ation are  found  in  Table  IV, 


Art.  91.] 


SHEARING  ALONG  NEUTRAL  SURFACE. 


573 


The  average  values  of  the  ultimate  shear  in  the  neutral 
surface  determined  by  Messrs.  Cline  and  Heim  in  their 
"  Tests  of  Structural  Timbers,"  already  cited,  are  given  in 
Table  V  for  nine  varieties  of  structural  timbers,  both  green 
and  air-seasoned.  These  results  belong  to  the  same  full- 
size  beams  as  the  values  given  in  Table  I  of  this  Article. 


TABLE   V. 

COMPUTED  SHEARING  STRESSES  DEVELOPED  IN  STRUCTURAL 

BEAMS 


Total 

First  Failure  by  Shear. 

Shear  Following  Other 

- 

Number 
of  Tests. 

Per  cent,  of  Total  and 
Average  per  Sq.  In. 

Failure. 
Per  cent,  of  Total  and 
Average  per  Sq.  In. 

Species. 

Green. 

Dry. 

Green. 

Dry. 

Green. 

Dry. 

% 

Lbs. 

%    • 

Lbs. 

% 

Lbs. 

% 

Lbs. 

Longleaf  pine  

17 

9 

54 

353 

56 

272 

23 

374 

O 

Douglas  fir  

IQI 

91 

2 

1  66 

6 

221 

22 

2QS 

49 

2Q4 

Shortleaf  pine  

48 

13 

17 

332 

46 

364 

6 

327 

8 

418 

Western  larch 

62 

c;2 

8 

288 

27 

^4O 

16 

^14. 

21 

^7O 

Loblolly  pine  

III 

25 

7 

335 

28 

434 

2 

**  f 

356 

16 

546 

Tamarack 

3O 

9 

10 

261 

33 

200 

1. 

26l 

o 

Western  hemlock  .  .  . 

39 

44 

5 

288 

23 

307 

28 

28l 

68 

438 

Redwood  

28 

12 

7 

302 

0 

II 

218 

17 

250 

Norway  pine  

49 

10 

6 

232 

IO 

278 

6 

266 

o 

It  will  be  observed  in  all  of  these  tests  that  there  is 
much  variation  in  the  intensities  of  the  different  stresses 
found  and  especially  in  these  ultimate  intensities  of  shear 
in  the  neutral  surfaces  of  full-size  beams.  As  has  already 
been  indicated  this  is  due  to  the  presence  of  a  variety  of 
weakening  defects  to  which  timber  is  subject.  This  sig- 
nifies that  low  working  stresses  s'hould  be  used. 

It  has  been  found  in  many  cases,  and  possibly  in  nearly 
all,  that  wind-shakes,  season  cracks,  and  other  influences 


574  BENDING  OR  FLEXURE.  [Ch.  XII. 

which  induce  at  least  partial  separation  of  the  fibres  at  the 
neutral  surface,  are  the  sources  of  incipient  failure  by  shear- 
ing in  the  neutral  surface. 

In  designing  timber  beams  this  liability  to  shear  along 
the  neutral  surface  should  always  be  carefully  tested  by 
computations.  Relatively  short  beams  are  particularly 
liable  to  fail  in  this  manner,  and  the  greater  part  of  the 
timber  beams  used  in  engineering  work  are  of  this 
class. 

It  is  a  very  simple  analytical  matter  to  establish  such 
a  relation  between  the  methods  of  failure  by  longitudinal 
shearing  and  rupture  of  the  fibres  as  to  indicate  more  or 
less  approximately  the  limit  beyond  which  one  mode  of 
failure  is  more  liable  to  occur  than  the  other,  but  empirical 
values  for  both  these  ultimate  resistances  have  been  seen 
to  be  so  variable  as  to  make  it  more  advisable  to  compute 
the  carrying  capacity  of  the  beam  by  both  methods,  especi- 
ally as  each  is  a  simple  procedure. 


Influence  of  Time  on  ike  Strains  of  Timber  Beams. 

It  has  been  found  by  actual  observation  that  if  a  timber 
beam  is  loaded  to  no  greater  extent  than  one  fourth  of  its 
ultimate  load,  the  resulting  deflection  will  continue  to  in- 
crease under  continued  loading  for  a  long  period  of  time. 
Sufficient  investigations  have  not  yet  been  made  to  express 
these  results  quantitatively  with  much  accuracy.  Enough 
has  been  ascertained,  however,  to  show  that  the  influence 
of  time  is  most  important  in  determining  the  deflection  of 
timber  beams  under  loads  applied  for  a  considerable  period 
of  time,  and  that  when  the  loading  becomes  a  large  portion 
of  the  ultimate,  i.e.,  perhaps  75  per  cent.,  the  beam  may 
fail  if  the  application  be  sufficiently  continued.  Indeed, 


Art.  91.]  CONCRETE  BEAMS.  575 

some  experiments  indicate  that  failure  may  possibly  take 
place  at  .6  or  .7  of  the  ultimate  of  a  single  application,  if 
that  amount  be  imposed  a  sufficient  length  of  time. 

It  should  be  understood,  therefore,  that  in  using  the  co- 
efficients of  elasticity  given  in  this  article  for  the  purpose 
of  computing  deflections,  such  computations  may  be  applic- 
able only  when  the  loads  are  applied  for  short  periods  of 
time. 

Concrete  Beams. 

When  a  concrete  or  a  natural  stone  beam  is  subjected  to 
transverse  loading  it  fails  by  tearing  apart  on  the  tension 
side.  The  failure  of  the  beams,  therefore,  indicates  to  some 
extent  the  ultimate  tensile  resistance  of  the  material.  Ob- 
viously, in  the  case  of  concrete  beams  the  ultimate  carrying 
capacity  will  depend  upon  a  number  of  elements,  such  as 
the  kind  and  quality  of  cement,  sand  and  broken  stone  used, 
and  the  proportions  of  the  mixture.  Table  VI  contains 
results  of  tests  of  a  considerable  number  of  concrete  beams 
6  ins.  by  6  ins.  in  cross-section  and  six  months  of  age.  For 
three  months  these  beams  were  frequently  wetted  though 
kept  in  air.  During  the  remaining  three  months  they  were 
kept  in  air  without  wetting.  The  length  of  span  for  some 
of  these  beams  was  42  ins.  and  18  ins.  for  the  remainder. 
Within  the  limits  of  the  tests  this  difference  in  span  appeared 
to  make  no  essential  difference  in  the  ultimate  intensities 
of  stress  in  the  extreme  fibres.  With  the  cross-sections  of 
the  beams,  i.e.,  6  ins.  wide  and  6  ins.  deep,  the  ratio  of  span 
length  divided  by  the  depth  was  either  7  or  3,  making  the 
beams  very  short.  The  different  columns  of  the  table  show 
the  character  of  the  ingredients  of  the  concrete  as  well  as 
the  greatest,  mean,  and  least  values  of  the  intensities  of  ex- 
treme fibre  stress  K.  As  would  be  anticipated,  the  values 


576 


BENDING  OR  FLEXURE. 


[Ch.  XII. 


TABLE   VI. 

CONCRETE  BEAMS  SIX  MONTHS  OLD. 


Concrete. 

Size  of 
Stone  in 
Inches. 

No.  of 
Tests. 

Ultimate  Stress  in 
Extreme  Fibres, 
Lbs.  per  Sq.  In. 

Max. 

Mean. 

Mm. 

B'klyn  Bridge  Rosendale  

.  s.  br. 
-2-4 

-3-5 
-2-4 

-3-5 
-2-4 

-3-5 
-2-4 

-3-5 

-2-4 

-3-5 
-2-4 

-3-5 
-2-4 

-3-5 
-2-4 

-3-5 
-2-4 

-3-5 
-2-4 

-3-5 
-2-4 

-3-5 
-2-4 

-3-5 
-2-4 

-3-5 
-2-4 

-3-5 
-2-4 

-3-5 
-2-4 

-3-5 

O-2* 
1-2* 
O-I 

°7,2* 
i7fi 

O-I 

O-2* 

Ijfi 

O-I 
0-2* 

Mi 

O-I 

Gravel. 

6 

5 
4 
3 
6 

3 
6 
6 
6 
6 
6 
6 

6 
6 
6 
6 

6 
6 
6 
6 
6 
6 
6 
i 
6 
6 
6 
6 

5 
6 

140 

128 

153 
140 

*?4 

1.8 
647 
516 
510 
458 
560 
516 
385 
329 
554 
297 

423 
329 
560 

49  1 
574 
460 

654 
54i 
192 

554 
417 
379 
279 
460 

373 

I03 
80 
128 
I36 
125 
126 
526 
449 
452 
402 

503 
420 

349 
283 
424 
268 

377 
272 
472 
404 

493 
419 
566 
484 
171 

157 
481 
352 
344 
245 
382 
3M 

76 

33 
109 

'34 
1  20 

122 
460 
360 

335 
360 
458 
355 
282 
238 
326 
224 
297 
238 
404 
34i 
453 
39i 
466 
414 
147 

414 
285 
312 

195 
326 
266 

<          « 

<          « 

<          <  < 

Alias  Portland 

< 

< 

< 

< 

Silica  Portland 

t 

i 

i 

i 

Alsen  Portland 

• 

« 

< 

B'klyn  Bridge  Rosendale  

Atlas  Portland   

Silica  Portland  

<  <           <  < 

Alsen  Portland     

" c. "  indicates  cement;   "s."  indicates  sand;    "br."  indicates  broken  stone 
or  gravel.     An  excellent  limestone  was  used  for  broken  stone. 

for  the  Portland  cement  beams  are  much  higher  than  those 
for  the  Rosendale  cement.  The  table  exhibits  the  usual 
variations  in  the  results  for  such  material,  but  on  the  whole 
those  for  gravel  are  seen  to  be  somewhat  less  than  those  for 
broken  stone,  the  proportions  of  mixture  being  the  same 


Art.  91.] 


CONCRETE  BEAMS. 


577 


for  the  two  materials.  Even  with  Portland  cement,  and 
with  as  rich  a  mixture  as  1-2-4,  the  results  show  that 
working  values  of  the  greatest  intensity  in  extreme  fibres 
should  not  exceed  40  to  60  pounds  per  square  inch. 

The  investigations  from  which  the  results  in  Table  VI 
have  been  taken  were  conducted  by  Messrs.  George  C. 
Saunders  and  Herbert  D.  Brown,  graduating  students  in  the 
class  of  Civil  Engineering  of  Columbia  University  in  1898. 

The  results  of  tests  of  twelve  Giant  Portland  cement 
concrete  beams  with  30-  and  6 8 -inch  spans  are  given  in 
the  "  U.  S.  Report  of  Tests  of  Metals  and  Other  Materials  " 
for  1900,  and  they  are  shown  in  Table  VII. 

TABLE    VII. 

TRANSVERSE  TESTS   OF  GIANT  PORTLAND-CEMENT 

CONCRETE  BEAMS. 
Composition:  i  c.,  3  s.,    5  br.  st. 


Span, 

Breadth, 

Depth, 

No.  of 

Ultimate  Stress,  k,  in   Extreme 
Fibres,  Pounds  per  Square  Inch. 

Max. 

Mean. 

Min. 

68 

6 

6 

I 

— 

472 



30 

6  and  4 

6 

7 

564 

493 

348 

30 

6  and  4 

6 

4 

454 

4i5 

367 

The  age  of  these  beams  was  made  up  of  2  days  in  air, 
2  months  in  water,  and  then  i  one  month  in  air,  making  a 
total  of  3  months  and  2  days.  The  broken  stone  included 
all  sizes  passing  a  2  J-inch  ring,  and  retained  on  a  sieve  with 
J-inch  meshes.  In  these  tests  the  ratio  of  length  over  depth 
was  5,  except  in  the  first  where  it  was  n.  There  seems  to 
be  little  difference  in  the  values  of  k  for  the  two  ratios,  but 
the  number  is  much  too  small  to  yield  any  law  of  variation. 

The  values  of  the  ultimate  extreme  fibre  stresses  k, 
shown  in  Table  VIII,  are  the  results  of  testing  to  failure 
short  Portland  cement  concrete  beams  by  Mr.  H.  Von  Schon, ! 


578 


BENDING  OR  FLEXURE. 


[Ch.  XII. 


Chief  Engineer  of  the  Michigan  Lake  Superior  Power  Com- 
pany, at  Sault  Ste.  Marie,  Mich.,  and  they  are  taken  from 
his  paper  in  the  "  Transactions  of  the  American  Society  of 
Civil  Engineers  "  for  December  1899.  The  beams  were  6 
inches  by  6  inches  in  cross-section,  with  a  span  of  18  inches. 
The  ratio  of  length  over  depth,  therefore,  was  3. 

TABLE    VIII. 

PORTLAND-CEMENT  CONCRETE  BEAMS,  6  INS.  BY  6  INS.  SECTION, 

18  INS.  SPAN. 


Cement. 

Broken  Stone. 

Mixture. 

No.  of 
Tests. 

Ultimate  Fibre  Stress,  K, 
Pounds  per  Square  Inch. 

Max. 

Mean. 

Min. 

E 

Sandstone 

A 

2 

I78 

I76 

174 

'  ' 

B 

2 

225 

217 

209 

<  < 

C 

2 

288 

280 

272 

<  « 

D 

2 

329 

325 

321 

'  ' 

E 

2 

1  08 

IO2 

97 

Boulder  stone 

A 

2 

354 

326 

298 

'  ' 

B 

2 

358 

328 

299 

" 

C 

2 

390 

373 

356 

*  * 

D 

2 

420 

410 

400 

'  ' 

E 

2 

350 

330 

310 

R 

Sandstone 

A 

2 

181 

169 

158 

«  « 

B 

2 

183 

i?5 

167 

<  < 

C 

2 

266 

262 

258 

<  « 

D 

2 

328 

308 

288 

<  < 

E 

2 

195 

182 

169 

Boulder  stone 

A 

2 

390 

347 

204 

'  ' 

B 

2 

423 

406 

390 

'  ' 

C 

2 

410 

392 

374 

'  ' 

D 

2 

411 

393 

375 

«  < 

E 

2 

332 

322 

312 

Mixture  A.  . .  . 
.     "        B 

C.  ... 

D 

E.  . 


cement,  2.4  sand,  5.3  broken  stone. 
2.4    "      4.8 
2.4    "      4.4      " 
2.4    "      4 
"       0.3  lime,  3.1  sand,  5.3  broken  stone. 


The  beams  were  left  from  two  to  eight  days  in  their 
forms  or  moulds  after  being  made,  and  then  tested  at  the 
age  of  60  days  in  air. 


Art.  91.] 


CONCRETE  BEAMS. 


579 


The  chief  elements  in. the  composition  of  the  Portland 
cements  indicated  by  E  and  R  in  the  Table  were  as  follows : 

Cement  E.  Cement  R. 

Lime 62 . 38  63 . 55 

Silica 23 . 08  2 1 . 70 

Alumina 5 . 69  8.76 

Magnesia 1.21  2.96 

Iron  oxide 5 . 35  1.27 

Potash  and  soda i  .66  1.12 

The  sand  used  in  Mr.  Von  Schon's  tests  was  from  St. 
Mary's  River,  the  broken  sandstone  was  the  native  Pots- 
dam variety,  while  the  broken  boulder  stone  was  granitic 
in  character.  All  broken  stone  would  pass  through  a  ij- 
inch  ring  and  be  retained  on  a  i-inch  ring;  the  material 
was,  therefore,  little  balanced. 

In  the  constructions  executed  under  the  supervision  of 
the  Boston  Transit  Commission,  large  amounts  of  concrete 
were  needed,  and  in  the  report  of  the  Commission  for  the 
year  ending  June  30,  1902,  there  are  exhibited  a  large  num- 
ber of  tests  of  Portland-cement  concrete  beams  6  inches 
by  6  inches  in  cross-section  with  30-inch  spans.  The  ratio 
of  length  of  span  over  depth  of  beam  in  this  case  is,  there- 
fore, 5.  Table  IX  gives  the  greatest,  average,  and  least 
results  of  these  tests  with  the  number  of  beams  broken. 

TABLE   IX. 

PORTLAND-CEMENT  CONCRETE  BEAMS,  6  INS.  BY  6  INS.  SECTION, 

30  INS.  SPAN. 


Composition  by 

Ultimate  Fibre  Stress,  k, 

Volume. 

Hours  in 

Air  Pressure, 

No  of 

Lbs.  per  Sq.  In. 

Compressed 
Air. 

Lbs.  per 
Sq.  In. 

Tests. 

Cement. 

Stone 
Dust.* 

Broken 
Stone.* 

Max. 

Mean. 

Min. 

I 

1-7 

2-75 

24 

7-12 

12 

999 

851 

677 

I 

1-9 

2.6 

24 

12-18 

50 

924 

850 

590 

I 

2 

2.4 

48 

18-25 

30 

904 

731 

622 

I 

2 

2.4 

28-30  days 

20-25 

IOO 

900 

728 

523 

.  *  Approximate  volumes. 


58o 


BENDING  OR  FLEXURE. 


[Ch.  XII. 


The  concrete  was»  machine  mixed  and  Vulcanite-Port- 
land cement  was  used.  The  stone  dust,  to  which  reference 
is  made  in  the  table,  was  finely  crushed  stone  varying  from 
impalpable  powder  up  to  J  inch  diameter,  the  broken  stone, 
on  the  other  hand,  being  of  ordinary  size.  It  will  be  noticed 
that  these  beams  were  kept  a  part  of  the  time  in  compressed 
air  at  pressures  varying  from  7  to  25  pounds,  presumably 
for  the  reason  that  some  of  the  material  was  to  be  used  under 
such  conditions. 

Table  X  contains  results  of  a  number  of  tests  of  con- 
crete beams  6  inches  by  6  inches  in  cross-section  and  with 
30-inch  spans,  made  for  the  purpose  of  comparing  the  re- 
sistances of  concretes  made  with  stone  dust  and  sand. 
This  table  is  also  taken  from  the  Report  of  the  Boston 
Transit  Commission  for  the  year  ending  June  30,  1902, 

TABLE   X. 

PORTLAND-CEMENT  CONCRETE  BEAMS,  6  INS.  BY  6  INS.  SECTION, 

30  INS.  SPAN. 


Composition  by  Volume  (Approximate). 

Ultimate  Fibre  Stress,  k, 
Pounds  per  Square  Inch. 

No.  of 

Tests. 

Cement. 

Sand. 

Stone 
Dust. 

Broken 
Stone. 

Max. 

Mean. 

Min. 

I 



2 

2.4 

4 

947 

848 

760 

I 

•  9 

•9 

2.7     • 

4 

846 

784 

704 

I 

1.6 



3 

4 

773 

711 

656 

I 

•  9 

•9 

2.7 

4 

862 

806 

759 

This  concrete  was  also  made  with  Vulcanite-Portland 
cement  and  the  mixing  was  done  by  hand.  The  beams 
were  kept  in  air  for  the  first  24  hours  and  then  29  days  in 
damp  earth. 

The  results  both  as  to  coefficient  of  elasticity  and  ex- 
treme fibre  stress,  given  in  Table  XI,  were  determined 
at  the  mechanical  laboratory  of  the  Department  of  Civil 


Art.  91.] 


CONCRETE  BEAMS. 


Engineering  of  Columbia  University  in  1902  by  Mr.  Myron  S. 
Falk.*  They  have  special  value  from  the  age  of  the  beams, 
which  was  about  seven  years.  These  beams  were  originally 
made  under  the  supervision  of  Mr.  A.  Black,  Instructor  in 
Civil  Engineering,  Columbia  University,  for  the  purpose  of 
determining  thermal  linear  expansion.  They  were  kept 
well  moistened  for  several  months  after  being  made,  but 
subsequently  until  tested  they  were  kept  under  cover  with- 
out moistening.  The  gravel  used  was  rounded,  varying  in 
size  from  -J  to  2^-  inches. 


TABLE   XI. 

PORTLAND-CEMENT  MORTAR  AND  CONCRETE    BEAMS  BROKEN 
BY  CENTRE  WEIGHT. 


Bar. 

Age, 
Years. 

Span  in 
Inches. 

Section  of  Bar 
in  Inches. 

Coefficient 
of  Elasticity, 
Pounds  per  Sq.  In. 

Extreme 
Fibre  Stress, 
Pounds  per 
Square  Inch 

Depth. 

Width. 

A 

7         A 

if> 

4T  ^ 

4.06 

A, 

•4 
7-4 

ou 
16 

.   I  4 

1,591,000 

278 

A, 

7-4 

16 

' 

'  ' 

I,IO2,OOO 

3*5 

B 

7 

36 

* 

4.00 

2,I22,OOO 

606 

B, 

7 

16 

' 

'  ' 

2,440,000 

636 

B2 

7 

16 

' 

'  ' 

1,220,000 

530 

C 

7 

36 

' 

4;p5 

1,315,000 

247 

% 

7 

16 

' 

387,000 

229 

C2 

7 

16 

' 

'  ' 

I,O23,OOO 

208 

D 

7-3 

36 

4.  10 

4;  15 

1,165,000 

294 

A 

7-3 

16 

' 

597,000 

415 

D2 

7-3 

16 

597,000 

346 

Bars  A,  i  Aalborg cement,  2  sand,  4  gravel. 
"    B,   i   Atlas  "        3     " 

"    C,    i  Alsen  "        3     "      5  gravel. 

"    D,  i       "  "2     " 

Some  of  the  coefficients  of  elasticity  are  abnormally 
low,  those  belonging  to  the  beams  B,  B^  and  B2  are  fairly 


*  See  Proc.  Am.  Soc.  C.  E.,  February,  1903. 


582  BENDING  OR  FLEXURE.  ]Ch.  XII. 

representative  of  what  may  be  expected  with  such  mate- 
rial in  flexure. 

Plate  A  represents  graphically  the  results  of  the  tests 
of  the  preceding  three  bars  B.  As  usual,  the  strain  of 
deflection  consisted  of  two  parts  in  all  cases,  one  perma- 
nent, at  least  for  the  time  being,  and  one  elastic,  which 
disappeared  on  the  removal  of  the  load.  This  feature  is 
shown  by  two  lines,  in  each  case  indicated  by  the  same 
letter  and  subscript.  The  difference  between  the  total 
and  permanent  strain  or  deflection  varied  very  nearly  as 
the  centre  load,  and  that  difference  being  the  elastic  de- 
flection was  used  in  computing  the  coefficients  of  elasticity 
given  in  Table  XI.  No  coefficient  of  elasticity  was  com- 
puted for  a  centre  loading  less  than  about  200  pounds 
For  the  purpose  of  computing  deflections  under  ordinary 
working  stresses  from  a  condition  of  little  or  no  loading, 
it  would  be  best  to  take  the  coefficient  of  elasticity  at  not 
more  than  one  half  of  the  values  given  in  the  Table,  in 
order  to  allow  for  that  part  of  the  deflection  which  does 
not  disappear  immediately  upon  the  removal  of  the  loading. 

Reviewing  all  the  preceding  values  of  the  ultimate 
stress  in  the  extreme  fibres  of  concrete  and  mortar  beams, 
the  working  intensities  of  stress  in  extreme  fibres  can  prob- 
ably not  be  properly  taken  higher  than  50  to  75  pounds 
per  square  inch  when  Portland  cement  is  used  for  well- 
balanced  mixtures  not  less  rich  than  i  cement,  2  sand,  and 
4  broken  stone,  or  possibly,  where  exceptionally  well  made, 
i  cement,  3  sand,  and  5  broken  stone.  If  gravel  is  em- 
ployed, some  reduction  should  be  made,  depending  upon 
its  character,  and  a  similar  observation  must  be  applied 
to  mixtures  less  rich  in  cement  than  the  preceding. 

For  natural  cements,  values  of  working  stress  greater 
than  one  fourth  of  the  preceding  probably  should  not  be 
used.  Indeed,  it  may  be  a  serious  question  whether 


Art.  91.] 


CONCRETE  BEAMS. 
PLATE  A. 


583 


.001 .002        .004 


DEFLECTION  AT  CENTERJN  INCHES 


584 


BENDING  OR  FLEXURE. 


[Ch.  XII. 


natural  cement  should  be  used  at  all  where  concrete  or 
mortar  may  be  subjected  to  flexure. 

TABLE    XII. 
BRICK-MASONRY  BEAMS. 
(Age  of  beams  about  equally  5  months,  8  days,  and  6  months.) 

ROSEND  ALE-CEMENT   MORTAR:      I    C.,    2   S. 


Span,  Inches. 

/ 
^  ' 

Stress  in  Extreme  Fibre,  Pounds  per 
Square  Inch. 

No.  of  Tests. 

Max. 

Mean. 

Min. 

96 

7-4 

67 

54 

38 

5 

78 

6 

— 

18 

66 

5-i 

81 

56 

23 

4 

42 

3-2 

9i 

73 

54 

8 

PORTLAND-CEMENT  MORTAR:    i  c.,  35. 


96 

7-4 

173 

144 

124 

4 

66 

5-i 

145 

1  20 

96 

4 

42 

3-2 

229 

1  66 

94 

10 

Table  XII  exhibits  some  interesting  results  of  the  tests 
of  brick-masonry  beams.  These  investigations  were  made 
by  Messrs.  A.  W.  Gill  and  Frederick  Coykendall,  gradua- 
ting students  in  Civil  Engineering  in  Columbia  University 
in  1897.  Fig.  i  shows  the  manner  of  laying  up  the  brick 
to  form  the  beams  which  were  tested.  The  breadth  of 
each  beam  was  about  12  ins.  and  the  depth  13  ins.  The 
spans  varied  from  8  ft.  down- to  3  ft.  6  ins.,  with  the 
ratios  of  length  over  depth  of  beam  given  in  the  column 

headed  -r.     This  column  of  ratios  shows  that  the  beams 
a 

should  be  considered  short. 

The  Rosendale-cement  mortar  was  mixed  with  one  vol- 


Art.  91.]  BRICK-MASONRY  BEAMS.  585 

ume  of  cement  to  two  volumes  of  sand,  while  the  Portland- 
cement  mortar  was  mixed  with  one  volume  of  cement  to 
three  volumes  of  sand.  During  the  first  three  months  the 
beams  were  kept  well  wetted,  but  less  so  during  the  last 
three  months.  At  no  time  were  they  dry.  The  Table  gives 


FIG.  i. 

all  the  results  of  tests  and  shows  that  the  beams  had  very 
little  resisting  capacity,  although  possibly  15  to  20  pounds 
per  square  inch  might  be  justified  as  working  values  in  the 
extreme  fibres  of  the  beams  built  with  Portland-cement 
mortar.  The  bricks  were  laid  by  ordinary  masons  with 
such  care  as  could  be  impressed  upon  them,  although  the 
experimenters  stated  that  the  brickwork  was  of  very  in- 
different quality  and  hence  that  the  results  are  lower  than 
they  should  be. 


586 


BENDING  OR  FLEXURE. 


[Ch.  XII. 


Natural-stone  Beams. 

Table  XIII  exhibits  results  found  by  the  same  experi- 
menters as  in  the  case  of  Table  XII  with  a  number  of. 
natural-stone  beams,  the  spans  for  which  varied  from  36 
ins.  down  to  12  ins.     The  first  figure  in  the  second  column1 
of  the  table  headed  "  Section  "  gives  the  depth  of  each  beam,  \ 

TABLE  XIII. 
NATURAL-STONE  BEAMS. 

BIvUESTONE. 


Stress  in  Extreme  Fibre,     ' 

Span, 

Section, 

/ 

Pounds  per  Square  Inch. 

No.  of 

—  . 

qv,otc 

a 

Max. 

Mean. 

Min. 

24 

4X6 

6.15 

3,958 

3,512 

3,054 

5 

36 

6X8 

6.2 

3,288 

2,797 

2,906 

3 

12 

4X6 

3 

4,112 

3,237 

2,282 

ii 

24 

8X6 

3 

3,929 

3,547 

2,715 

6 

GRANITE. 


24 

4X6 

6 

2,321 

2,250 

2,178 

3 

36 

6X8 

6 

1,861 

i,798 

1,766 

3 

12 

4X6 

3 

2,714 

2,487 

2,086 

9 

SANDSTONE. 


24 

4X6 

6 

i,575 

i,354 

i,237 

3 

36 

6X4 

6 

1,204 

945 

637 

3 

12 

4X6 

3 

1,907 

i,539 

1,267 

9 

MARBLE. 


24 

4X6 

6 

2,036 

1,886 

1,617 

3 

36 

6X8 

6 

1,683 

i,548 

i,354 

3 

12 

4X6 

3 

2,455 

2,026 

1,696 

9 

Art.  9i.]  NATURAL-STONE  BEAMS.  587 

while  the  second  figure  gives  the  width.     It  will  be  observed 

/ 
from  the  ratios  of  -j  given  in  the  third  column  that  the  beams 

were  very  short.  The  extreme  fibre  stresses  are  seen  to  run 
comparatively  high  for  the  bluestone,  granite,  and  marble. 
Indeed,  working  values  of  intensities  may  reasonably  be 
taken  as  follows: 

For  blue  tone 250  to  400  pounds  per  square  inch. 

"    granite. 200  to  300       " 

"    marbl   17    10225       " 

"    sandstone iootoi5o       "         "       " 

In  the  use  of  sandstone  it  should  be  understood  that  the 
preceding  values  apply  only  to  the  best  qualities  of  that 
particular  stone. 


CHAPTER   X1IL 
,«fwi;.  CONCRETE-STEEL  MEMBERS. 

Art.  92.— Composite  Beams  or  Other  Members  of  Concrete 

and  Steel. 

CONCRETE,  like  other  masonry,  is  admirably  adapted  to 
resist  compression.  Its  capacity  of  resistance  to  tension 
is  much  less  than  its  ultimate  compressive  resistance, 
although  if  the  concrete  is  well  made  the  tensile  resistance 
may  have  considerable  value.  The  purpose  of  the  con- 
crete-steel combination  is  the  production  of  a  beam  or 
other  member  almost  entirely  of  concrete,  but  which  shall 
have  a  high  capacity  to  resist  tension  in  those  portions 
which  may  be  subjected  to  tensile  stresses.  This  result 
is  accomplished  by  embedding  steel  bars  of  desired  shape 
and  of  suitable  cross -sectional  area  in  the  proper  parts  of 
the  concrete.  While  no  general  rule  can  be  given  for  the 
area  of  the  steel  section  in  comparison  with  the  concrete, 
it  may  be  stated  approximately  that  the  steel  section  is 
usually  between  f  and  i  J  per  cent,  of  the  area  of  a  normal 
section  of  the  concrete.  Inasmuch  as  the  presence  of  the 
steel  is  for  the  purpose  of  giving  tensile  resistance  to  the 
member  it  is  evident  that  the  re-enforcing  steel  bars  will 
always  be  found  in  those  portions  of  the  concrete  mass 
which  may  be  subjected  to  tension.  In  such  concrete- 
steel  construction  as  arches  the  steel  re-enforcement  is 
frequently  used  both  on  the  tension  and  compression  sides 
of  the  concrete. 

588 


Art.  93.]  CONCRETE-STEEL   BEAMS.  589 

In  the  case  of  concrete-steel  beams  or  other  similar 
members,  as  the  steel  is  entirely  embedded  in  the  concrete, 
the  loads  and  reactions  must  obviously  be  applied  directly 
to  the  latter.  When  the  concrete  takes  its  stress,  there- 
fore, at  least  a  portion  of  that  stress  must  be  conveyed  to 
the  steel,  and  that  requires  that  the  adhesive  joint  or  bond 
between  the  steel  and  concrete  shall  be  as  strong  as  possible. 
Hence  in  laying  the  steel  bars  in  the  concrete  it  is  necessary 
that  the  contact  between  the  two  materials  shall  be  inti- 
mate and  essentially  continuous.  Various  means  are  em- 
ployed to  accomplish  these  ends.  Square  bars  are  fre- 
quently twisted,  while  round  bars  may  be  nicked  and  flat 
ones  either  twisted  continuously  in  one  direction  or  have 
alternate  portions  twisted  in  opposite  directions,  or,  finally, 
rolled  with  alternately  enlarged  and  contracted  sections. 
Again,  where  built-up  members  are  embedded  in  concrete, 
rivet-heads  and  other  details  of  construction  serve  the 
same  general  purposes.  The  efficiency  of  the  concrete- 
steel  construction  depends  wholly  upon  the  resistance  of 
this  bond,  and  the  design  must  always  be  such  that  the 
adhesive  shear,  so  to  speak,  or  the  stress  of  sliding  along 
the  steel  surface,  shall  never  exceed  per  square  unit  the 
ultimate  resistance  of  the  bond. 

In  the  analysis  and  computations  which  follow  it  is 
assumed,  as  it  must  be,  that  the  bond  between  the  steel 
and  concrete  is  such  as  to  make  the  entire  mass  act  as  a 
unit,  so  that  the  combination  of  the  two  heterogeneous 
elements  shall  act  as  a  single  whole. 

Art.  93.— Physical  Features  of  the  Concrete-steel  Combination 

in  Beams. 

It  will  be  shown  later  on  that  so  far  as  can  be  deter- 
mined from  physical  data  now  available  the  coefficient  of 
elasticity  for  concrete  in  compression  for  the  operations 


590  CONCRETE-STEEL   MEMBERS.  [Ch.  XIII. 

ordinarily  employed  in  designing  engineering  structures 
^and  for  mixtures  not  less  rich  in  cement  than  i  cement, 
3  sand,  and  6  gravel  or  broken  stone,  at  ages  of  one  to  six 
months,  may  range  from  about  2,000,000  pounds  per  square 
inch  to  more  than  4,000,000  pounds  per  square  inch,  while 
for  concrete  beams  the  coefficient  or  modulus  may  range 
from  about  1,500,000  pounds  per  square  inch  for  compara- 
tively shallow  beams  to  more  than  3,000,000  pounds  per 
square  inch  for  beams  of  comparatively  great  depths. 
Values  for  the  coefficient  of  elasticity  for  concrete  in 
tension  can  be  found  in  Art.  60.  Further  tests  for  the 
determination  of  this  quantity  are  much  to  be  desired,  but 
enough  has  been  done  to  establish  at  least  closely  approxi- 
mate values.  Some  authorities  assume  the  tensile  coeffi- 
cient to  be  much  less  than  the  coefficient  of  elasticity  for 
concrete  or  mortar  in  compression.  As  a  matter  of  fact, 
the  tests  of  a  Momer  arch  of  75  feet  span  by  a  committee 
of  the  Austrian  Society  of  Engineers  and  Architects, 
which  made  its  report  in  1895,  showed  in  that  particular 
case  the  coefficient  of  elasticity  of  concrete  in  tension  to  be 
nearly  one  fifth  greater  than  the  coefficient  for  compression, 
although  it  should  be  stated  that  the  age  of  the  tensile 
specimens  was  materially  greater  than  that  of  the  com- 
pression material.  The  values  in  Art.  60  indicate  that  the 
tensile  coefficient  is  at  least  equal  to  the  compressive. 
It  is  possible  that  subsequent  investigations  may  show 
that  the  tensile  coefficient  of  elasticity  is  less  than  that  for 
compression,  but  at  the  present  time  there  appears  to  be 
practically  no  basis  for  that  assumption.  It  seems  to 
be  reasonable  and  safe,  as  it  is  more  simple  to  take 
the  two  coefficients  equal  to  each  other  until  further 
investigations  have  conclusively  established  a  different 
ratio. 

It  is  important  to  state  in  this  connection  that  the  re- 


Art.  93-]  CONCRETE-STEEL   COMBINATION   IN  BEAMS.  591 

suits  of  tests  with  concrete-steel  beams,  so  far  as  they  have 
been  made,  indicate  that  the  elastic  or  semi-elastic  behavior 
of  concrete  under  stress  will  in  the  main  characterize  the 
behavior  of  the  same  material  when  under  loading  in  the 
composite  beam  of  concrete  and  steel,  so  that  the  coefficients 
of  elasticity  determined  for  concrete  alone  may  be  used  in 
the  composite  member. 

There  is  one  important  respect  in  which  the  action  of 
concrete  alone  is  quite  different  from  that  which  takes  place 
when  it  is  combined  with  steel.  In  the  latter  case  the  con- 
crete will  stretch  under  a  stress  nearly  or  quite  equal  to  its 
ultimate  resistance  a  comparatively  large  amount.  It  is 
sometimes  stated  that  under  such  conditions  the  coeffi- 
cient of  tensile  elasticity  of  the  concrete  is  practically  zero, 
but  there  is  just  as  much  ground,  or  more,  for  making  the 
same  observation  in  connection  with  such  ductile  materials 
as  structural  steel.  What  is  actually  meant  is  simply  that 
the  concrete  will  stretch  before  parting  much  more  when  its 
deformation  is  controlled  by  the  corresponding  deformation 
of  the  steel  reinforcement  than  when  it  acts  by  itself  or 
without  such  reinforcement.  This  feature  of  the  action 
under  stress  of  concrete  in  the  composite  beam  has  a  most 
important  bearing  upon  some  rather  peculiar  phenomena 
connected  with  the  testing  of  such  beams  to  failure.  M. 
Considere  has  stated  (' '  Comptes  Rendus  Academic  des  Sci- 
ences, ' '  Paris,  Dec.  12,  1898)  that  mortar  will  stretch  twenty 
times  as  much  when  combined  with  steel  as  when  unaided 
by  that  combination.  He  further  states  that  the  concrete 
stretches  uniformly  with  uniform  increments  of  bending 
moment  up  to  about  four  tenths  of  the  ultimate  moment. 

As  the  coefficient  of  elasticity  for  concrete  is  a  small 
fraction  only  of  that  of  steel  the  tendency  of  the  concrete 
in  composite  beams  is  to  stretch  or  compress  more  than 
the  steel  embedded  in  it.  Hence  the  concrete  immediately 


592 


CONCRETE-STEEL   MEMBERS. 


[Ch.  XIII. 


adjacent  to  the  steel  tends  to  slide  along  the  latter,  but 
that  tendency  is  resisted  ^by  the  adhesive  shear  at  the  joint, 
in  consequence  of  which  the  steel  acquires  its  stress  whether 
of  tension  or  compression.  The  normal  section  of  the 
unloaded  beam,  therefore,  will  not  remain  normal  after 
flexure,  but  there  will  be  either  a  cup-shaped  depression 
around  the  steel  or  a  similar  shaped  elevation.  This  is 
illustrated  in  Fig.  i. 

A 


JcTi 


FIG.  i. 

In  that  figure  the  intensity  of  stress  on  either  side  of 
the  neutral  axis  is  assumed  to  vary  directly  as  the  distance 
from  the  axis,  but  in  a  subsequent  analysis  a  different  law  of 
variation  will  be  assumed  in  order  that  the  treatment  may 
be  complete,  although  the  author  is  not  of  opinion  that  the 
assumption  of  any  law  of  variation  different  from  that  of 
the  common  theory  of  flexure  is  at  the  present  time  justified. 
It  will  further  be  assumed  in  the  analysis  which  follows 
that  normal  sections  of  the  unloaded  beam  will  remain 
normal  under  loading.  This  is  a  common  procedure,  and 
it  is  not  believed  that  the  amount  of  variation  from  a  plane 
section  under  stress,  described  above,  is  sufficient  to  make 
the  assumption  sensibly  in  error. 

Art.  94 — Rate  at  Which  Steel  Reinforcement  Acquires  Stress. 

The  determination  of  the  rate  at  which  the  concrete 
gives  stress  to  the  steel  is  not  of  great  importance  in  ordi- 
nary design  work  or  in  most  other  practical  relations;  yet 


Art.  94-]  RATE  OF  ACQUIRING  STRESS.  593 

it  is  desirable  in  some  cases,  and  it  is  an  element  of  the 
action  of  internal  stresses  in  a  composite  beam  which 
should  be  understood  as  clearly  as  practicable.  The  fol- 
lowing analysis  offers  a  means  of  determining  that  rate 
as  nearly  as  it  can  be  done  at  the  present  time.  The 
notation  used  is  shown  also  in  Fig.  3  on  the  opposite  page. 
The  intensity  of  stress  in  the  concrete  at  the  distance 
dv  the  distance  of  the  steel  reinforcement,  from  the  neutral 
axis  is  k.  Then  if  /  represent  the  moment  of  inertia  of  the 
entire  composite  section  about  its  neutral  axis  (located  by 
dlt  determined  hereafter),  there  may  be  written 


If  5  is  the  total  transverse  shear  in  the  normal  section 
in  question  at  the  distance  x  from  one  end  of  the  beam, 

dk  J 


(2) 


Let  p  be  the  total  perimeter  of  section  of  the  steel  re- 

inforcement at  the  section  located  by  x. 
Let  A  2  be  the  area  of  steel  section  with  perimeter  p. 
Let  s'be  the  intensity  of  adhesive  shear  at  the  surface  or 

joint  between  the  steel  and  concrete. 
Let  k2  be  the  intensity  of  stress  in  the  steel. 

The  variation  of  k2  for  the  indefinitely  small  distance 
dx  is  dk2.     From  what  has  preceded  there  may  be  written 

.     .     .     (3) 


.2        ..     . 

Inserting  the  value  of  dx  from  eq.  (3)  in  eq.  (2), 

dk      dk 

: 

, 


, 

O/i,      :  =~r~/  . 

d 


594  CONCRETE-STEEL   MEMBERS.  [Ch.  XIII. 

By  solving  this  equation  for  s'  and  remembering  that 

dk2=E2 
dk  ~Ei 

,_<-./,  d2  dk2  _  ^>E2  d2  A  2  /  \ 

This  value  of  s'  must  never  exceed  the  ultimate  adhe- 
sive resistance  between  the  steel  and  concrete. 

Tests  for  the  determination  of  the  adhesive  shear  between 
concrete  and  imbedded  round  rods  have  been  made  by  Pro- 
fessors Talbot,  Withey,  Hatt,  Duff  A.  Abrams  and  others. 
In  view  of  the  inevitable  uncertainties  of  condition  of  such 
rods  in  respect  to  the  bond  between  them  and  the  concrete, 
greatly  varying  values  must  be  anticipated,  as  they  will 
depend  upon  the  age  proportions  of  the  concrete,  the  smooth- 
ness (or  roughness)  of  the  surface  of  the  rods,  the  amount 
of  water  used  in  mixing  the  concrete  and  the  continuity  of 
contact  between  the  concrete  and  the  rods.  The  value  of 
adhesive  shear  has  sometimes  been  taken  as  16  to  20  per 
cent,  of  the  ultimate  compressive  resistance  of  the  concrete, 
but  this  is  probably  too  high,  even  for  the  best  qualities  of 
concrete. 

Again,  the  ultimate  value  of  adhesive  shear  as  deter- 
mined by  the  pulling  of  rods  directly  from  a  block  of  con- 
crete may  be  materially  different  from  that  developed  in 
a  bent  beam  and,  hence,  the  latter  procedure  should  be  the 
basis  of  determinations  for  reinforcing  rods  for  beams.  A 
clear  distinction  should  be  drawn  between  the  adhesive 
shear  existing  prior  to  movement  of  the  rod  in  its  mastic 
and  the  resistance  to  that  motion  after  it  once  begins. 

Professor  M.  O.  Withey  published  in  a  Bulletin  of  the 
University  of  Wisconsin,  No.  321,  1909,  the  data  of  a  large 
number  of  tests  in  which  the  results  were  obtained  from 


Art.  94.]  ADHESIVE  SHEAR  OR  BOND.  595 

loaded  beams,  the  stretch  of  the  rods  being  accurately 
measured  by  an  extensometer  for  a  given  length  of  imbedded 
rod.  The  diameter  of  rod  was  f  inch  and  the  age  of  the 
concrete  varied  from  seven  days  up  to  six  months.  A  large 
number  of  tests  gave  the  adhesive  shear  as  varying  from 
a  minimum  of  129  pounds  per  square  inch  to  a  maximum 
of  362  pounds,  a  few  only  of  the  results  falling  below  200 
pounds  per  square  inch.  It  would  probably  be  fair  to 
take  250  pounds  per  square  inch  as  a  representative 
average  of  these  results. 

In  a  series  of  tests  with  diameters  of  bars  running  from 
f  inch  to  i  inch,  the  average  results  were  278  and  286 
pounds  per  square  inch  for  the  two  smaller  sizes  of  bars 
and  163  pounds  and  195  pounds  per  square  inch  for  the 
i -inch  bars.  The  age  of  the  1-2-4  concrete  in  this  case 
was  two  months. 

There  may  be  found  in  Bulletin  No.  71,  University  of 
Illinois,  a  full  account  of  a  large  number  of  "  Tests  of  Bond 
between  Concrete  and  Steel,"  by  Duff  A.  Abrams.  These 
tests  were  made  under  a  great  variety  of  conditions  as  to 
age,  sizes  of  rods,  surface  of  rods,  i.e.,  whether  plain  or 
deformed,  shapes  of  cross-sections,  rods  pulled  out  of  blocks 
and  rods  stressed  in  reinforced  concrete  beams,  accompanied 
by  extended  observations  as  to  effects  of  loading  including 
careful  measurements  of  the  stretch  of  steel  both  in  pulling 
rods  from  blocks  and  as  they  were  stressed  in  beams.  In 
these  tests  a  clear  distinction  was  recognized  between  the 
adhesion  to  the  surface  of  the  rods  and  the  resistance  of 
movement  after  initial  slip,  the  greatest  intensity  of  bond 
resistance  usually  being  developed  after  the  beginning  of 
slip. 

A  roughened  surface  of  rod  will  obviously  yield  a  greater 
bond  resistance  than  a  perfectly  smooth  surface,  the  resist- 
ance of  the  latter  being  almost  wholly  adhesion. 


596 


CONCRETE-STEEL   MEMBERS. 


[Ch.  XIII. 


The  following  are  a  few  of  Mr.  Abrams'  conclusions: 
"  (41)  The  mean  computed  values  for  bond  stresses  in 
the  6 -foot  beams  in  the  series  of  1911  and  1912  were  as 
given  below.  All  beams  were  of  1-2-4  concrete,  tested  at 
2  to  8  months  by  loads  applied  at  the  one  third  points  of 
the  span.  Stresses  are  given  in  pounds  per  square  inch. 


Number 
of  Tests. 

First  End 
Slip  of  Bar. 

End  Slip 
of  o.ooi  In. 

Maximum 
Bond  Stress 

i  and  i  f-in.  plain  round  

28 

245 

140 

775 

f-in.  plain  round  

T. 

1  86 

242 

274 

f-in.  plain  round    

7 

172 

2^5 

255 

i  -in  plain  square 

6 

IQO 

248 

278 

i  -in.  twisted  square  .' 

T. 

222 

289 

717 

I  f-in.  corrugated  round  

9 

251 

360 

488 

"  (42)  In  the  beams  reinforced  with  plain  bars  end  slip 
begins  at  67  per  cent,  of  the  maximum  bond  resistance; 
for  the  corrugated  rounds  this  ratio  is  51  per  cent.,  and  for 
the  twisted  squares,  66  per  cent. 

"  (43)  The  bond  unit  resistance  in  beams  reinforced 
with  plain  square  bars,  computed  on  the  superficial  area  of 
the  bar,  was  about  75  per  cent,  of  that  for  similar  beams 
reinforced  with  plain  round  bars  of  similar  size. 

"  (44)  Beams  reinforced  with  twisted  square  bars  gave 
values  at  small  slips  about  85  per  cent,  of  those  found  for 
plain  rounds.  At  the  maximum  load,  the  bond-unit  stress 
with  the  twisted  bars  was  90  per  cent,  of  that  with  plain 
round  bars  of  similar  size. 

"  (45)  In  the  beams  reinforced  with  if -inch  corrugated 
rounds,  slip  of  the  end  of  the  bar  was  observed  at  about 
the  same  bond  stress  as  in  the  plain  bars  of  comparable 
size.  At  an  end  slip  of  o.ooi  inch,  the  corrugated  bars  gave 
a  bond  resistance  about  6  per  cent,  higher  and  at  the  maxi- 
mum load,  about  30  per  cent,  higher  than  the  plain  rounds. 


Art.  94.]  ADHESIVE  SHEAR  OR  BOND.  597 

"  (46)  The  beams  in  which  the  longitudinal  reinforce- 
ment consisted  of  three  or  four  bars  smaller  than  those  used 
in  most  of  the  tests  gave  bond  stresses  which,  according  to 
the  usual  method  of  computation,  were  about  70  per  cent, 
of  the  stresses  obtained  in  the  beams  reinforced  with  a  sin- 
gle bar  of  large  size." 

As  the  greatest  bond  stress  was  developed  after  the 
beginning  of  slip,  the  preceding  results  show  that  such  a 
maximum  value  exists  beyond  a  net  slip  of  o.ooi  inch. 

Again  referring  to  the  resistance  of  deformed  bars,  he 
states,  "  The  mean  bond  resistance  for  the  deformed  bars, 
tested  was  not  materially  different  from  that  for  plain 
bars  until  a  slip  of  about  .01  inch  was  developed;  with  a 
continuation  of  slip,  the  projections  came  into  action  and 
with  much  larger  slip  high  bond  stresses  were  developed." 

Again  referring  to  a  working  bond  stress,  he  states : 

"  (59)  A  working  bond  stress  equal  to  4  per  cent,  of  ths 
compressive  strength  of  the  concrete  tested  in  the  form  of 
8-  by  1 6 -inch  cylinders  at  the  age  of  28  days  (equivalent  to 
80  pounds  per  square  inch  in  concrete  having  a  compressive 
strength  of  2000  pounds  per  square  inch)  is  as  high  a  stress 
as  should  be  used.  This  stress  is  equivalent  to  about  one 
third  that  causing  first  slip  of  bar  and  one  fifth  of  the  maxi- 
mum bond  resistance  of  plain  round  bars  as  determined 
from  pull-out  tests.  The  use  of  deformed  bars  of  proper 
design  may  be  expected  'to  guard  against  local  deficiencies 
in  bond  resistance  due  to  poor  workmanship  and  their 
presence  may  properly  be  considered  as  an  additional  safe- 
guard against  ultimate  failure  by  bond.  However,  it  does 
not  seem  wise  to  place  the  working  bond  stress  for  deformed 
bars  higher  than  that  used  for  plain  bars." 

The  preceding  results  were  obtained  from  statically 
loaded  beams.  Professor  Withey  found  no  injurious  effects 
on  the  resistance  of  adhesive  shear  under  repeated  loads  until 


59$  CONCRETE-STEEL   MEMBERS.  [Ch.  XIII. 

the  latter  became  50  to  60  per  cent,  of  the  ultimate  static 
loads.  This  last  percentage  may  be  raised  to  60  to  70  per 
cent,  with  corrugated  bars.  Investigations  made  by  the 
same  authority  indicate  that  the  results  of  static  tests  on 
smooth  round  rods  imbedded  in  beams  will  give  values 
for  the  bond  or  adhesive  shear  between  the  concrete  and 
the  rods  from  one  half  to  two  thirds  only  of  corresponding 
results  obtained  by  pulling  imbedded  steel  rods  from  the  con- 
crete cylinders,  but  Mr.  Abrams  appears  to  believe  that 
the  results  of  properly  made  ' '  pull-out ' '  tests  will  be  about 
the  same  as  found  for  beams. 

While  materially  larger  values  for  ultimate  resistance 
of  adhesive  shear  have  been  reported  by  some  experimenters 
with  small  rods,  it  appears  prudent  not  to  take  the  ultimate 
resistance  greater  than  perhaps  200  to  350  pounds  per  square 
inch  for  round  or  square  rods  from  ij  inch  to  f  inch  in 
diameter. 

The  working  value  for  this  bond  for  adhesive  shear 
should  not  be  taken  more  than  one  fourth  to  one  fifth  of 
its  ultimate  value. 


Art.   95.— Ultimate   and  Working  Values   of  Empirical   Quan- 
tities for  Concrete-steel  Beams. 

It  is  necessary  for  the  practical  use  of  the  preceding 
and  following  analyses  that  a  number  of  empirical  quanti- 
ties be  determined,  chiefly  for  the  concrete.  The  coeffi- 
cient of  elasticity  for  wrought  iron  for  this  purpose  may 
be  taken  at  28,000,000  pounds  per  square  inch,  and 
30,000,000  pounds  per  square  inch  for  structural  steel, 
which  is  now  generally  used  in  the  reinforcement  of  con- 
crete-steel beams. 

The  modulus  of  elasticity  for  concrete  at  different  ages 
and  for  different  proportions  of  matrix  and  aggregate  has 


Art.  95.]  ULTIMATE  AND  WORKING  VALUES.  599 

been  fully  considered  in  Art.  67,  and  Table  I  of  that  Article 
exhibits  a  full  set  of  values.  A  mixture  of  i  cement,  2  sand 
and  4  broken  stone  or  gravel  is  generally  used  in  rein- 
forced concrete  work ;  and  for  such  concrete  the  Table  cited 
above  shows  that  the  modulus  of  elasticity  at  the  age  of 
one  month  may  be  taken  from  about  1,500,000  to  nearly 
3,000,000.  In  view,  however,  of  the  uncertain  conditions 
attending  the  making  of  concrete  on  actual  work  a  higher 
value  than  2,000,000  is  seldom  used.  The  ratio  of  the 
modulus  for  steel  divided  by  that  for  concrete  is  generally 
taken  at  15,  although  12  is  sometimes  employed,  the  latter 
value  implying  a  modulus  for  concrete  of  2,500,000. 

The  ultimate  resistances  of  mortar  and  concrete  in 
tension  and  compression  will  be  found  in  Arts.  60  and  67. 
These  values  will  also  depend  upon  the  proportions  and 
character  of  mixture  or  upon  the  age.  The  records  of 
tests  and  experience  which  have  thus  far  accumulated  in 
connection  with  concrete-steel  construction  show  that  the 
compressive  working  stress  of  concrete  in  beams,  where  the 
mixture  is  in  the  proportions  of  i  cement,  2  sand,  and  4 
gravel  or  broken  stone,  may  probably  be  taken  as  high  as 
500  pounds  per  square  inch.  It  should  be  remembered 
that  this  intensity  will  exist  in  the  extreme  fibres  of  the 
beam  only.  Mixtures  of  less  strength  would  require  a 
corresponding  reduction  in  the  maximum  working  in- 
tensity of  compression.  A  mixture,  for  example,  of  i 
cement,  2^  sand,  and  5  broken  stone,  unless  the  materials 
were  well  balanced,  might  justify  a  reduction  of  the 
greatest  working  stress  to  400  pounds  per  square  inch. 

Some  foreign  authorities  have  prescribed  two  degrees 
of  safety,  in  the  first  of  which  the  maximum  working  stress 
of  compression  of  427  pounds  per  square  inch  is  allowed, 
and  711  pounds  per  square  inch  for  safety  of  the  second 
degree.  Structures  in  which  the  duty  of  the  concrete  is 


6oo  CONCRETE-STEEL  MEMBERS.  [Ch.  XIII. 

severe  might  be  designed  with  the  smallest  of  those  values, 
but  where  the  duty  is  materially  less  severe,  with  the 
larger. 

It  is  not  unusual  at  the  present  time  in  the  design  of 
concrete-steel  arches  to  allow  a  maximum  intensity  of 
compression  of  500  pounds  per  square  inch  and  50  to  75 
pounds  per  square  inch  for  the  maximum  intensity  of 
tension,  if  tension  is  allowed. 

Tensile  tests  of  concrete  show  that  where  proportions 
of  i  cement,  2  sand,  and  4  gravel  or  broken  stone  are  used 
a  maximum  intensity  of.  tension  of  50  to  70  pounds  per 
square  inch  is  about  J  to  J  the  ultimate  tensile  resistance 
at  the  age  of  three  to  six  months.  These  values  are  reason- 
able and  may  be  employed  in  concrete  work  where  it  is 
permitted  to  avail  of  the  tensile  resistance  of  concrete.  In 
much  of  the  best  engineering  practice  at  the  present  time, 
however,  the  tensile  resistance  of  the  concrete  is  neglected 
in  the  interests  of  additional  safety  in  concrete-steel  beam 
construction.  Inasmuch  as  fine  cracks  may  appear  in 
concrete  from  other  agencies  than  tensile  stress,  it  is  un- 
doubtedly advisable  in  most  cases  certainly  to  omit  the 
bending  resistance  of  the  concrete  in  tension,  especially 
as  that  omission  does  not  sensibly  increase  the  weight  or 
cost  of  the  beam  when  properly  designed. 

Art.  96. — General  Formulae  and  Notation  for  the  Theory  of 
Concrete-steel  Beams  According  to  the  Common  Theory  of 
Flexure. 

The  application  of  the  common  theory  of  flexure  to 
the  bending  of  concrete-steel  beams  is  in  reality  the  de- 
velopment of  the  theory  of  flexure  for  composite  beams 
of  any  two  materials.  The  notation  to  be  used  and  the 
general  formulas  will  first  be  written,  therefore,  and  then 
the  special  formulae  for  concrete-steel  beams  will  be  estab- 


Art.  96.]  GENERAL   FORMULA  AND  NOTATION.  601 

lished  in  the  succeeding  articles.  These  general  formulae, 
it  should  be  observed,  apply  to  beams  of  any  shapes  of 
cross-section  of  either  material  or  for  any  relative  areas 
of  cross-section  of  those  materials,  although  in  concrete- 
steel  beams  the  area  of  cross-section  of  the  steel  is  frequently 
or  perhaps  usually  but  one  to  one  and  a  half  per  cent,  of 
the  area  of  the  concrete. 

Again,  the  formulae  will  be  so  written  as  to  make 
practicable  the  use  of  different  coefficients  of  elasticity 
for  concrete  in  tension  and  compression  if  that  should 
be  desired. 

The  notation  to  be  used  in  the  succeeding  articles  is 
chiefly  the  following: 
E2  =  coefficient  of  elasticity  of  the  steel. 
El  =         "          "          "         "     "    concrete  in  compression. 
nEi  =         "          "          "         "     "    concrete  in  tension. 
Al  and  A2  are  the  areas  of  normal  section  of  the  concrete 

and  steel  respectively. 

7j  and  /2  are  the  moments  of  inertia  of  Al  and  A2  respec- 
tively about  the  neutral  axis  of  the  normal  section. 
kl  =  greatest  intensity  of  bending  compression  in  the  con- 
crete. 

kr  =  greatest  intensity  of  bending  tension  in  the  concrete. 
c  =  greatest  intensity  of  bending  compression  in  the  steel. 
t  =  greatest  intensity  of  bending  tension  in  the  steel. 
b=  breadth  of  the  concrete. 

h  and  hj  are  total  depths  of  the  concrete  and  steel  re- 
spectively. 
h2=  vertical    distance  between  the    centres  of  the  steel 

reinforcing  members. 

d^  —  distance  of  extreme  compression  "fibre"  of  the  con- 
crete from  the  neutral  axis. 

d2  =  distance  of  the  centre  of  the  compression  steel  rein- 
forcing member  from  the  neutral  axis. 


602  CONCRETE-STEEL  BEAMS.  [Ch.  XIII. 

ds  =  distance  from  the  neutral  axis  to  the  centre  of 

the  tension  steel  reinforcement. 
dj  =  distance  from  extreme  compression  fibre  of  the 

steel  to  the  neutral  axis. 

a  =  distance  of  the  centre  of  the  compression  steel 
reinforcing  member  from  exterior  compression 
surface  of  concrete. 

a  i  =  distance  of  the  centre  of  the  tension  steel  rein- 
forcing   member    from    exterior    tension    sur- 
face of  concrete. 
M2=area  of  normal  section  of  reinforcing  steel  in 

tension. 
(i—  r) .A 2=  area   of   normal   section   of  reinforcing   steel  in 

compression. 
k=  intensity  of  corrrpressive  stress  in  the  concrete 

at  distance  z  from  the  neutral  axis. 
k"  =  intensity  of  tensile  stress  in  the  concrete  at  dis- 
tance z  from  the  neutral  axis. 
£2  =  intensity  of  stress  in  the  steel  at  distance  z  from 

the  neutral  axis. 

u  =  tensile  or  compressive  strain  in  unit  length  of 
" fibre"  at  unit  distance  from  the  neutral 
axis. 

In  all  the  theory*  of  bending  of  concrete-steel  beams  it 
is  assumed,  as  in  the  common  theory  of  flexure,  that  any 
plane,  normal  section  of  the  beam,  before  bending  takes 
place,  will  remain  plane  (and  normal)  while  the  beam  is 
subjected  to  bending.  Hence 

k=EiUZ,     k"  =nE\uz,     and     k2=E2uz.       .     (i) 

Inasmuch  as  all  the  loading  carried  by  concrete-steel 
beams  is  supposed  to  act  in  a  direction  normal  to  the  axes 

*  Given  in  Art.  32.     Eqs.  (i)  to  (4)  are  simple  adaptations  of  the  equa- 
tions of  that  Art.  to  this  case. 


Art.  96.] 


GENERAL  FORMULA  AND  NOTATION. 


603 


of  the  beams,  as  is  usual  in  the  common  theory  of  flexure, 
the  total  stresses  of  tension  and  compression  in  any  normal 
section  of  a  beam  induced  by  the  bending  must  be  equal 
to  zero.  The  expression  of  this  sum,  written  by  the  aid  of 
eqs.  (i)  and  by  which  the  neutral  axis  of  the  composite 
section  is  determined,  is  the  following: 


Or 


/v,  ro  £    /w 

/    zdA.  +  n  zdAt  +  j?  I      lt£dAt~o. 

J  0  Jhi—di  tLif  dz'—hj 


(3) 


i 

q 

T" 

( 

I, 

i 

! 

1 

• 

' 

t         i 

hlti 

i 

•*-Q—  GL—  0*- 

! 

Eq.  (3)  is  perfectly  general,  and  the  position  of  the 
neutral  axis  can  always  be  located  by  it  whatever  may 

be  the  shape  of  cross-section 
of  either  the  concrete  or  steel. 
Fig.  i  may  be  taken  as 
an  arbitrary  typical  com- 
'  posite  section  showing  the 
preceding  system  of  notation 
applied  to  it.  The  outline 
of  the  concrete  is  rectangular, 
as  in  the  ordinary  concrete-steel  beam.  The  steel  in  com- 
pression is  represented  as  two  steel  angles,  while  three 
round  rods  constitute  the'  steel  in  tension.  In  the  next 
article  the  application  of  the  general  eq.  (3)  to  the  special 
case  of  the  ordinary  concrete-steel  beam  will  be  made. 

The  general  value  of  the  bending  moment  of  the  stresses 
induced  in  any  normal  section  of  a  composite  beam  can  be 
at  once  written  by  the  aid  of  eqs.  (i).  The  typical  ex- 
pression of  the  differential  moment  is 


FIG.  i. 


604  CONCRETE-STEEL  BEAMS. 

Hence  the  value  of  the  moment  is 


[Ch.  XIII. 


IAV     (4) 


This  equation  is  also  completely  general  whatever  may 
be  the  shape  of  section  of  either  material.  It  will  be  de- 
veloped for  the  ordinary  form  of  concrete-steel  beams  in 
Art.  97. 

Eqs.  (3)  and  (4)  cover  completely  the  theory  of  bending 
or  flexure  of  composite  beams  of  two  materials,  one  of 
them  having  different  values  for  the  coefficients  of  elasticity 
in  tension  and  compression.  It  will  be  observed  that  the 
position  of  the  neutral  axis  of  any  section  of  the  beam,  as 
located  by  eq.  (3),  is  affected  by  the  values  of  Ev  E2,  and  ny 
and  that  it  does  not  in  general  pass  through  the  centre  of 
gravity  of  the  section. 

Art.  97. — T-Beams  of  Reinforced  Concrete. 

The  general  formulae  of  Art.  96  belong  to  beams  of  any 
shape  of  cross-section  whatever;  it  is  only  necessary,  there- 


FIG.  i. 


fore,  in  this  case,  to  apply  them  to  the  T-shaped  section. 
Two  conditions  may  arise,  in  one  of  which  the  neutral 


Art.  97.]  T-BEAMS  OF  REINFORCED  CONCRETE.  605 

axis  lies  in  the  flange  of  the  beam  whose  cross-section  is 
shown  in  Fig.  i,  or,  as  shown  in  that  figure,  it  may  lie  below 
the  flange.  As  is  usually  the  case  in  actual  work,  the 
tensile  resistance  of  the  concrete  will  finally  be  neglected. 
This  latter  condition  makes  it  necessary  to  consider  only 
the  case  shown  by  Fig.  i . 


Position  of  Neutral  Axis. 

Using  the  notation  of  Art.  96  under  the  conditions  out- 
lined above,  but  first  recognizing  the  tensile  resistance  of 
the  concrete, 


Again, 


/**  rdi  rdi-f 

I    zdAi=  I      z-bldz+  I        z-bdz 

JO  Jdi-f  JO 

-,/(*  -f 

zdAi 

-dl 


As  the  steel  section  is  small  it  will  be  essentially  correct 
to  consider  each  part  of  it  concentrated  at  its  centre  of 
gravity.  Hence  there  may  be  written, 

/•</.' 


-d2')  =A2(d2-rh2).     (ib) 


Introducing  the  values  given  by  eqs.  (i),  (ia)  and  (2) 
in  eq.  (3)  of  Art  96, 


606  CONCRETE-STEEL  BEAMS.  [Ch.  XIII. 

7ixj      7 1/2     bd2        7  ,  ,       bf2      d2bn         ,,    ,  bh* 

o  fa.  —o  — [-  — l baj  +  — —  — — l h  non.a.  —  n  — 1— 

22  22  2 


i  —  n  i  —  n 


El  b       i  —  n 

The  solution  of  this  quadratic  equation  will  give 
'b1 


i  — 


i-n  Bi  6     i-w  (i-w)2 

3 

If  the  two  coefficients  of  elasticity  for  concrete  in  tension 
and  compression  are  the  same,  as  is  always  assumed  in 
actual  work,  n  =  i.  This  value  gives  indetermination  in 
Eq.  3,  but  it  is  only  necessary  to  multiply  both  members 
of  Eq.  2  by  (i  —  n)  and  then  make  n  =  i.  These  opera- 
tions give 


1         /  «•>_  \  £•>        .       'I  1          .      -C">      -A- 


d, 


If  the  entire  steel  reinforcement  is  on  the  tension  side  of 
the  beam,  r  =  i,  and  in  Eqs.  3  and  4,  a  +  r/z2  =a+/*2=^i 
The  tensile  capacity  of  the  concrete  is  practically  always 


Art.  97.]  T-BEAMS    OF  REINFORCED  CONCRETE.  607 

neglected ;  hence  n  =  o  in  Eq.  3,  and 


These  formulae  locate  the  neutral  axis  by  giving  the  dis- 
tance Jj  for  all  cases. 

An  important  special  case  arises  where  the  neutral  axis 
NS,  Fig.  i,  lies  in  the  lower  side  of  the  flange,  i.e.,  when 
di  =/.  Making  that  substitution  in  the  equation  preceding 
eq.  (2), 


,  <>bf  —nb  ,7/77     .  E2  A  \     nbhi2  .  E2  A    ,         1  ^      ,,N 
di2  --  \-di(nbhi  +  —  A2)  =  -  -+^-A2(a  +  rh2).     (6) 
2  \  ki\      I         2         rL\ 

Solving  this  quadratic  equation, 


If  concrete  in  tension  be  neglected,  n  =o  and, 


£2^2^    I/E2A2\2  ,    E2A2(a 

—        p;      77~  rt*  /  I  ~      T~T  ]    ~\~2—  —. 

Ei  b       \vEi  bf  /        Ei          b' 


Eq.  (8)  shows  that  the  case  of  a  T-beam  with  neutral 
axis  at  the  lower  surface  of  the  flange  and  with  tensile 
resistance  of  concrete  neglected  is  equivalent  to  a  solid 
rectangular  beam  of  the  same  width  as  the  flange  under 


6o8  CONCRETE-STEEL   BEAMS.  [Ch.  XIII. 

the  same  assumption  of  the  neglect  of  the  concrete  in  ten- 
sion. No  material  error  will  be  committed  in  assuming 
any  T-beam  similarly  equivalent  to  a  solid  rectangular 
beam  if  the  neutral  axis  is  near  the  under  side  of  the  flange. 
If  the  neutral  axis  NS  lies  in  the  flange  the  area  (bf  —  b) 
(f—di)  of  concrete  flange  section  will  be  in  tension.  In 

that  case  the  term  —n(b'  —b)—  -  --  must  be  added  to  the 

2 

third  member  of  eq.  (ia),  and  hence  to  the  first  member  of 
the  equation  preceding  eq.  (2).  This  will  add  obvious  cor- 
responding terms  to  eqs.  (3),  (4)  and  (5),  but  the  special 
case  is  so  rare  that  it  needs  no  further  attention.  Unless 
(f—di)  has  material  value  eqs.  (7)  and  (8)  may  be  used. 

Balanced  or  Economic  Steel  Reinforcement. 

In  order  that  there  may'  be  economy  of  material  it  is 
necessary  that  the  relation  between  the  cross-sectional  areas 
of  the  steel  and  concrete  may  be  such  as  to  make  the 
greatest  intensities  of  stress  in  each  equal  to  the  prescribed 
working  stresses.  This  condition  is  said  to  make  a  "  bal- 
anced "  section  or  a  balanced  percentage  of  steel  reinforce- 
ment. 

In  the  general  case  of  tensile  and  compressive  steel 
reinforcement  with  the  tensile  resistance  of  concrete  recog- 
nized, the  equality  of  the  total  tensile  and  compressive 
stresses  in  a  normal  section  of  a  T-beam  gives  eq.  (9),  if 
the  neutral  axis  lies  in  the  under  surface  of  the  flange,  as 
is  assumed  in  establishing  eqs.  (6),  (7)  and  (8); 


(9) 


Adding  \k\\)'d%  to  each  side  of  eq.  (9)  and  then  dividing  the 
resulting  equation  by  b'(di+dz)  =b'hi,  eq.  (10)  will  result: 


Art.  97.]  T-BEAMS   OF  REINFORCED  CONCRETE.  609 


or 


(xoa) 


It  will  now  be  convenient  to  simplify  the  forms  of  the 
preceding  equations  by  using  the  following  notation  : 

iff 

e=—,  the  ratio  of  the  modulus  of  elasticity  for  steel 
Ei 

over  that  for  concrete.     Usually  0  =  15,  but  occa- 
sionally 0  =  12. 

p=—-^-,  the  steel  ratio,  usually  expressed  as  per  cent,  of 

b  hi 

total  rectangular  section,  i.e.,  in  'case  of  the  T-beam 
per  cent,  of  total  rectangular  outline  b'h\. 


The  steel  ratio  or  per  cent,  p,  is  written  in  terms  of  the 
circumscribing  rectangle  b'hi  in  the  interests  of  simplicity 
and  as  being  at  least  as  rational  as  any  other  method. 

The  effective  depth  of  the  beam  is  taken  as  hi  because 
the  exterior  thickness  of  concrete  (h—hi)  is  usually  a  pro- 
tecting shell  against  fire,  possibly  to  be  partially  or  wholly 
destroyed  in  a  conflagration,  and,  hence,  not  to  be  counted 
as  effective  beam  material.  The  formulae  may  easily  be 
changed  so  as  to  be  expressed  in  terms  of  the  full  .depth 
h  by  simply  writing  h—  o  for  hi,  o  being  the  difference 
(h—hi),  i.e.,  the  thickness  of  the  concrete  from  the  centre 
of  the  tension  steel  reinforcement  to  the  lower  surface  of 
the  web  or  stem  of  the  beam,  usually  2  to  3  inches,  or 


6io  •        CONCRETE-STEEL    BEAMS.  [Ch.  XIII. 

more  for  very  large  beams.     The  preceding  notation  will 
enable  the  following  formulae  for  practical  use  to  be  written. 

Formula  to  Locate  Neutral  Axis  in  T-Beams. 

T-^-   •  f  /  \  1      7         j       -j.-      E>2  b  A.2  r      £2  A.2 

Dividing  eq.  (3)  by  hi  and  wntmg  —  —  -rj-  for  — -  — ; 

£Li  0     0  rL,i    0 

bf 


7  7       V  L  /      '  '     L 

di  =    =     hi\b       /  o 

hi  i  —  n 


V      \/2 

T--j)r*+n     u/  /        1 

b       /hi2  b'  ^  a+rh2      \hi\b 


(    •     \2  • 

»  i-n  b      (i-n)hi  (i-n)2 

Doing  precisely  the  same  with  eq.  (4)  there  will  result 
for  the  usual  condition  of  the  two  moduli  for  tension  and 
compression  being  the  same,  but  with  tensile  resistance  of 
the  concrete  recognized  : 


For  the  special  case  of  neglect  of  the  tensile  resistance 
of  concrete,  eq.  (5)  gives,  after  dividing  both  sides  by  hi : 


J2  ^r/74-r/7.  /  f  /h'          \         h'       \2 


y 

If  the  neutral  axis  lies  in  the  under  side  of  the  flange, 


Art.  97-1  T-BEAMS  OF  REINFORCED   CONCRETE.  611 

both  sides  of  eq.  (7)  are  to  be  divided  by  hi,  and  that  equa- 
tion may  then  take  the  form : 


Or,  if  concrete  in  tension  be  neglected,  n  =o  and  eq.  (8) 
then  gives 


If  the  reinforcing  steel  is  wholly  on  the  tension  side  of 
the  beam  section  r  =  i  and  a+rh2=a+h<2=h\.     Hence  in 

eqs.  (12),  (13),  (14)  and  (15),-^  —  -  =  i  ,  but  no  other  change 

hi 

is  needed. 

The  value  of  the  steel  ratio  or  per  cent,  for  the  perfectly 

general  case  is  given  by  eq.  (10),  by  placing  p  =TT^-  in  that 

b  hi 

equation  and  then  solving  for  p,  which  will  give: 


c(i-r)-rt      ..... 
If  concrete  in  tension  be  neglected,  eq.  (9)  shows  that 

k  i 

—dz=ki=o  in  eq.  (16),  and  that  equation  will  then  take 
the  form 


2C(i-r)-rt         2  (c(i  -r)  -rt)hi 


612  CONCRETE-STEEL  MEMBERS.  [Ch.  XIII 

If  the  reinforcing  steel  is  wholly  on  the  tension  side  of 
the  beam  section,  r  =  i  and  c(i  —  r)  —  rt  —  —  rt.  Eq.  (17) 
will  then  take  the  form  : 


, 


The  laws  of  the  common  theory  of  flexure  give  the  fol- 
lowing relations : 

ki     t    i  d\     eki  f     N 

T=~T>   or   T=-T>  .    .    .    •    (19) 

a\     e  0,3  0,3       t 

hence : 

eki+t     hi  f     , 


d* 
Also: 


-,     and     i=C 


Placing  the  value  of  r-^  from  eq.  (20)  in  eq.  (18) : 
n\ 


The  area  of  the  steel  section  A  2  can  at  once  be  found  from 
p  in  all  cases  by  simply  writing  : 

p=77r-,  and  hence,    A2=pb'hi.       .     .     (23) 


In  all  these  equations  for  locating  the  neutral  axis  of  a 

section  NS,  Fig.  i  ,  the  ratios  —  ,  ~-  and  other  similar  quan- 

o    h\ 


Art.  97.]  T-BEAMS  OF  REINFORCED   CONCRETE.  613 

titles  depending  on  the  dimensions  of  the  cross-section  will 
be  known,  at  least  tentatively.  Indeed  in  making  prac- 
tical applications  of  these  equations  it  will  in  general  be 
necessary  to  assign  trial  dimensions  of  the  cross-sections  of 
the  beam  if  the  application  is  made  for  the  design  of  the 
latter.  Such  trial  dimensions  must  be  assigned  by  the  aid 
of  prior  experience  or  other  beams  already  designed  for 
more  or  less  similar  conditions.  After  trial  dimensions  have 
been  tested  by  actual  computations  for  the  assigned  loads, 
such  modifications  or  revision  of  these  dimensions  as  may 
be  necessary  must  then  be  made. 

If  the  neutral  axis  lies  within  the  section  of  the  flange, 
the  changes  in  the  preceding  formulas  already  indicated  for 
that  case  may  be  easily  introduced,  but  the  case  is  so  rare 
that  complete  expressions  for  its  treatment  need  not  be 
.written.  If  the  tensile  resistance  of  the  concrete  is  neg- 
lected, the  formulas  for  the  special  case,  only,  of  the  neutral 
axis  lying  in  the  under  surface  of  the  flange  are  needed, 
simply  considering  the  depth  of  flange  /  as  the  distance  from 
the  upper  flange  surface  to  the  neutral  axis.  In  fact  that 
special  case  will  cover  the  great  majority  of  T-beams  with 
sufficient  accuracy  for  practical  purposes. 

The  general  value  of  the  steel  ratio  or  per  cent,  for  a 
balanced  section  may  be  considered  as  given  by  eq.  (16) 
even  though  the  neutral  axis  does  not  lie  in  the  under  sur- 
face of  the  flange,  at  least  as  a  reasonably  close  approxima- 
tion even  when  the  position  of  the  neutral  axis  is  materially 
different  from  that  supposition.  In  determining  that  ratio 
or  per  cent.  k\,  c  and  t  must  be  considered  as  prescribed 
working  values  of  those  respective  intensities  of  stress,  the 
ratio  between  c  and  t  being  fixed  by  the  distances  of 
the  steel  reinforcements  from  the  neutral  surface.  When 
the  steel  reinforcement  is  wholly  on  the  tension  side,  as 
in  the  usual  cases,  ki  and  t  are  prescribed  working  stresses 


614  CONCRETE-STEEL   MEMBERS.  [Ch.  IXII. 

for  the  concrete  in  compression  and  the  steel  in  tension, 
respectively. 

Art.    98.  —  Bending    Moments    in    Concrete-steel   T-Beams   by 
Common  Theory  of  Flexure. 

The  complete  expressions  for  the  bending  moments  of 
concrete-steel  T-beams  may  now  be  written  and  their 
values  for  any  particular  case  estimated,  by  introducing 
the  notation  already  employed  into  eq.  (4)  of  Art.  95. 
The  moment  of  inertia  or  integral  in  the  last  term  of  the 
second  member  of  that  equation  takes  the  form  : 


z2dA2  =  (i  -r)A2d22+rA2d32.  (i) 


Referring  to  Fig.  i  of  Art.  96,  the  other  two  moments 
of  inertia  in  the  first  and  second  terms  of  the  second  member 
of  eq.  (4)  of  Art.  95  become:  '**** 

dl?dAl=Vd^_(b,_fy(di-/)3f  xx 

o  3  3 


Jhi 


......     (3) 


'hi  -di  3 

Also, 

77               kl                  ,  K              E2z7               #2&1                              f    v 

Eiu=—\    and  E2u=—E1u=——.     .     .     (4) 


Introducing  these  values  in  eq.  (4)  of  Art.  95,  remem- 
bering that  hi—di=h2—d2=d3  the  bending  moment  M  for 
a  T-beam  become : 


(s) 


Art.  98.]  BENDING  MOMENTS  IN  CONCRETE-STEEL    T-BEAMS.    615 

This  equation  is  written  in  terms  of  one  intensity  of  stress 
k\  for  convenience  in  computation,  but  it  will  be  advisable 
sometimes  to  use  other  intensities,  such  as  the  greatest 
.  stress  t  in  the  tensile  steel  reinforcement.  This  can  readily 
be  done  by  the  aid  of  the  following  relations  based  upon 
common  theory  of  flexure,  in  addition  to  the  relations 

shown  in  eq.  (4)  and  remembering  that  -=r=e. 


kr     ki  ,,v 

=..     .     .     (6) 


These  simple  values  will  enable  any  intensity  to  be  expressed 
in  terms  of  any  other.  The  greatest  compression  in  the 
concrete,  k\,  and  greatest  tension  in  the  steel,  t,  are  those 
mostly  required. 

If,  as  is  usual,  the  two  moduli  of  elasticity  of  concrete  in 
tension  and  compression  are  equal  to  each  other,  n  =  i  in  eq. 
(5),  but  no  other  change  is  needed. 

/  - 
Neglect  of  Concrete  in  Tension. 

If  the  resistance  to  concrete  in  tension  be  neglected, 
n=o  in  eq.  (5),  and: 


.  (7) 


In  ordinary  T-beams  all  steel  reinforcement  is  in  tension; 
hence  r  =  i  and  eq.  (7)  becomes: 


6i6  CONCRETE-STEEL  MEMBERS.  [Ch.  XIII. 

Special  Case  of  Neutral  Axis  in  under  Surface  of  Flange. 
In  this  case  (d\  —  /)  =o  and  eq.  (7)  will  take  the  form: 


If  the  steel  reinforcement  is  wholly  on  the  tension  side  r  =  i  , 
as  in  eq.  (8). 

This  special  case  may,  without  material  error,  be  con- 
sidered to  include  all  T-beams  for  which  (d\  —  /)  or  (f—di) 
is  relatively  small. 

Art.  99.  —  Concrete  Steel  Beams  of  Rectangular  Section. 

All  formulae  for  reinforced  concrete  beams  with  rect- 
angular section  may  be  written  at  once  from  those  for 
T-beams  by  simply  making  bf  =  b  in  the  latter.  A  typical 
rectangular  cross-section  for  the  general  case  is  shown  in 
Fig.  i,  Art.  96,  although  in  the  usual  case  the  steel  rein- 
forcement is  wholly  on  the  tension  side. 

Formula  to  Locate  Neutral  Axis  in  Beams  of  Rectangular 

Section. 

The  general  case  requires  eq.  (n)  of  Art.  97.  Making 
b'  =  6  that  equation  becomes: 


—  =q= ±\/~~    — \-2ep-, —     -T-: — h~7 t^r.     (i) 

hi  i—n      \i—  n  (i —n)hi      (i —n)2 


If  the  moduli  for  tension  and  compression  are  the  same, 
as  is  invariably  assumed  in  engineering  practice,  b  =  bf  in 
eq.  (12),  Art.  97: 

a+rh2 


, 

-  +ep 
2  h 


Art.  99.]  CONCRETE-STEEL  BEAMS  OF  RECTANGULAR  SECTION.     617 

If  the  tensile  resistance  of  the  concrete  be  neglected, 
the  same  substitution  of  b  =b'  is  made  in  eq.  (13)  of  Art.  97  : 


.     .     (3) 


When  the  reinforcing  steel  is  wholly  on  the  tension  side 
r  =  i  and  a+rh2  =  a+h2  =h\,  hence: 

-±=q  =  -ep±V2ep+e2p2.   ....     (4) 

hi 

This  is  the  ordinary  case. 

It  will  be  observed  that  eq.  (3)  is  identical  with  eq.  (15) 
of  Art.  97. 

The  steel  ratio  or  per  cent.,  p  =7-^-,  for  the  general  case 

ok  i 

of  balanced  sections  is  given  by  eq.    (16)   of  Art.   97  by 
making  b  =  b'  : 


When  the  tensile  resistance  of  the  concrete  is  neglected, 
the  value  of  p  given  by  eq.  (17)  of  Art.  9*7  remains  un- 
changed : 


Eq.  (18)  of  the  same  article  gives  p  as  it  stands  if  the 
tensile  resistance  of  the  concrete  is  neglected,  i.e.,  of  r  =  i  : 


6i8  CONCRETE-STEEL   MEMBERS.  [Ch.  XIII. 

Eqs.  (19),  (20)  and  (21)  of  Art.  97  hold  true  for  rect- 
angular sections  and,  hence,  eq.  (7)  may  take  the  form  of 
eq.  (22)  of  that  article: 


These  values  of  the  steel  ratio  p  will  form  the  basis  of 
economical  beam  design.  The  working  stresses  k\  for  con- 
crete in  compression  and  t  for  steel  in  tension  will  be  pre- 
scribed in  the  specifications  for  the  work  to  be  done. 

Bending  Moments  for  Rectangular  Sections. 

The  general  value  of  the  bending  moment,  M,  i.e.,  for 
unequal  moduli  in  tension  and  compression  and  with  tensile 
resistance  of  concrete  recognized,  is  given  by  eq.  (5)  of  Art. 
98  after  making  bf  =  b. 


(9) 


3 


If  it  be  desired  to  express  this  equation  in  terms  of  other 
intensities  than  k\y  the  following  relations  given  in  eq.  (6) 
of  Art.  98  will  enable  that  to  be  done : 

eki      t       c         A    k'     k\  xx 

— :— =  — =  -r-    and    —  =  — .  do) 


The  moduli  for  concrete  in  tension  and  compression  are 
invariably  considered  equal,  and  in  that  case  n  =  i  in  eq.  (9) , 
but  no  other  change  is  required. 


Art.  99-J    CONCRETE-STEEL  BEAMS  OF  RECTANGULAR  SECTION.    619 

Neglect  of  Concrete  in  Tension. 

The  neglect  of  the  concrete  in  tension  is  affected  by 
making  n  =  o  in  eq.  (9)  giving : 


M=-^    — —  +eA2(d22(i  —  r) -\-d%2r)   .     .     .     (n) 


The  steel  reinforcement  is  usually  wholly  on  the  tension 
side,  i.e.,  r  =  i.     Making  this  substitution  in  eq.  (n)  : 

(12) 


All  the  preceding  values  of  the  bending  moment  M  may, 
if  desired,  be  expressed  in  terms  of  the  steel  ratio  p  by  sub- 
stituting pbhi  for  A  2. 

In  the  preceding  equations  the  distance  d\  of  the  neutral 
axis  from  the  exterior  compression  surface  of  the  beam  is 
to  be  found  from  the  appropriate  formula  for  q  of  this  article, 
since  d\  =  qh\. 

The  preceding  equations  complete  all  that  are  necessary 
in  the  treatment  of  practical  questions  of  design  or  of 
ultimate  carrying  capacity. 

In  all  the  preceding  analyses  of  Arts.  (97),  (98)  and  (99), 
the  total  depth  h  of  either  the  T-beam  or  the  solid  rect- 

A 

angular  section  may  be  used  if  desired  by  making  p  =  yyr- 

b  h 
£ 

or  =  —  >  but  in  that  case  in  the  equations  for  d\  the  fraction 
oh 


a+h2 


(when  r  —  i)  will  occur,  that  fraction  having  values 


varying  from  about  .67  for  floor  slabs  to  .95,  for  beams  of 

much  depth  instead  of  — — -  =  i .     It  is  rare,  however,  that 

»i 

such  a  form  of  equation  will  need  to  be  used. 


620 


CONCRETE-STEEL  MEMBERS. 


[Ch.  XIII. 


Art.   100. — Shearing  Stresses  and  Web  Reinforcements  in 

Reinforced  Concrete  Beams. 

• 

In  the  case  of  reinforced  concrete  T-beams  it  will  be 
assumed  that  the  stem  or  web  extending  from  the  upper 
surface  of  the  flange  down  to  the  centre  of  the  tension  steel, 
i.e.,  having  the  depth  hi  and  the  width  ft,  will  carry  the 
whole  transverse  shear.  In  the  solid  rectangular  section, 
the  total  sectional  area  less  that  part  of  it  below  or  outside 
of  the  centre  of  the  tension  steel  reinforcement  will  be 
assumed  to  resist  transverse  shear,  i.e.,  the  resisting  area 
will  be  bhi  as  in  the  case  of  the  T-beam. 

Fig.  i  represents  a  simple  T-beam  supported  at  each 
end  Q  and  R,  having  steel  reinforcement  both  in  the  flange 


6  a 


L         M 


FIG.  i. 

and  in  the  lower  or  tension  part  of  the  beam.  In  order 
to  illustrate  fully  the  action  of  the  shearing  stresses  in  such 
a  beam,  the  tensile  resistance  of  the  concrete  may  be  recog- 
nized. If  there  were  no  steel  reinforcement,  the  analysis 
of  Art.  15  shows  that  in  the  case  of  a  rectangular  section 
the  greatest  intensity  of  either  longitudinal  or  transverse 
shear  exists  at  the  neutral  axis  of  the  section  and  has  the 
value  of  f  the  average  shear  on  the  whole  section.  If  s 
be  that  maximum  intensity  of  shear  and  if.  5  is  the  total 
external  transverse  shear  at  the  given  section,  then  will 


Art.  ioo.]     SHEARING  STRESSES  AND  WEB  REINFORCEMENTS.     621 

s  =  -rr-.     In  Fig.    i,   Oc  =  0a=s  and  both  of    the  curves 

3  wfci 

Ae'a  and  Bee  are  parabolas  with  the  vertices  at  a  and  c, 
so  that  horizontal  ordinates  from  AO  to  the  curve  in  the 
one  case  and  from  BO  to  the  curve  in  the  other  case  repre- 
sent intensities  of  the  longitudinal  and  transverse  shear  at 
the  points  from  which  those  horizontal  ordinates  are  drawn. 
This  is  the  condition  of  the  shearing  stresses  in  beams  of 
a  single  material  subjected -to  flexure,  and  reinforced  con- 
crete beams  represent  simitar  members,  but  of  two  materi- 
als. The  stresses  given  to  the  longitudinal  steel  reinforce- 
ments may  be  assumed  provisionally  to  be  conveyed  to  them 
from  the  neutral  surface  at  a  constant  intensity  s\  and  in 
Fig.  i  that  constant  intensity  is  represented  by  cd  and  ab. 
The  curves  df  and  bf  are  drawn  so  as  to  make  a  constant 
horizontal  ordinate  between  them  and  the  parabolas  already 
indicated.  The  total  maximum  intensity  of  longitudinal  or 
transverse  shear  at  the  neutral  axis  will,  therefore,  be  the 
sum  of  5  and  si\  this  may  be  taken  with  sufficiently  close 
approximation,  at  least  for  practical  purposes,  as  f  the 
total  average  transverse  shear  at  a  given  section.*  Even 
if  the  horizontal  ordinate  between  the  two  curves  is  not 
uniform,  this  value  of  the  maximum  intensity  may  properly 
be  used. 

In  the  case  of  the  tensile  resistance  of  the  concrete 

*  It  has  come  to  be  the  practice,  for  some  reason  not  easily  appreciated, 
to  treat  the  transverse  shear  in  the  normal  section  of  a  reinforced  concrete 
beam  as  if  it  were  uniformly  distributed  over  that  normal  section,  which 
is  an  error  on  the  side  of  danger.  In  the  interests  of  both  safety  and  cor- 
rect analysis,  the  established  variation  of  intensity  of  shear  in  the  normal 
section  of  a  bent  beam  should  be  recognized,  for  it  holds  just  as  much  for 
a  resisting  concrete  section  as  for  a  section  of  any  other  material.  When 
the  bending  resistance  of  the  concrete  on  the  tension  side  is  ignored,  the 
law  of  variation  of  intensity  will  change,  but  the  maximum  intensity  at  the 
neutral  axis  will  be  unchanged. 


622 


CONCRETE-STEEL  MEMBERS. 


[Ch.  XIII. 


being  neglected,  Fig.  2,  representing  a  part  of  a  continuous 
reinforced  concrete  T-beam,  shows  the  variation  of  the  in- 
tensity of  the  longitudinal  and  transverse  shears.  The 
parabolic  curve  A  a  shows  the  variation  of  the  intensity  of 
the  shear  in  passing  from  the  neutral  axis  0  of  the  section 
to  the  exterior  surface  A,  aO  being  the  maximum  intensity 
and  equal  to  f  the  average  intensity  for  the  entire  section. 
Inasmuch  as  the  tensile  resistance  of  the  concrete  is  neg- 
lected, the  maximum  intensity  of  longitudinal  shear  Ob  =  Oa 
may  be  considered  as  varying  by  some  unknown  law  such, 
however,  as  to  make  the  total  internal  transverse  shear 


0  = 


FIG.  2. 

equal  to  the  total  external,  cd  representing  the  intensity 
at  the  centre  of  the  tension  steel  reinforcement. 

It  is  impossible  to  analyze  with  complete  accuracy  the 
variation  of  the  intensity  of  shear  in  the'  concrete  by  which 
the  reinforcing  steel,  either  in  tension  or  compression,  ac- 
quires its  stress,  but  it  cannot  have  a  uniform  value  equal 
to  the  maximium  intensity  at  the  neutral  surface.  It  is  to 
be  understood  that  the  constant  horizontal  shear  ordinates 
in  both  Figs,  i  and  2  are  to  be  interpreted  in  this 
manner. 

The  shearing  resistance  of  concrete  in  any  plain  or 
reinforced  concrete  structure  is  of  uncertain  value,  much 
as  is  the  tensile  resistance,  although  the  practice  of  crediting 


Art.  ioo.]  SHEARING  STRESSES  AND   WEB  REINFORCEMENTS.      623 

it  with  some  material  amount  may  be  justified.  At  the 
same  time  the  incipient  surface  cracks  which  are  found  to 
form  with  lapse  of  time  at  any  point  may  extend  deep 
enough  ultimately  to  prejudice  seriously  resistance  to  shear. 
It  is  probably  hazardous,  therefore,  to  depend  upon  concrete 
alone  to  resist  transverse  shear  in  beams,  either  of  the  T 
form  or  solid  rectangular,  or,  of  any  other  form.  In  fact 
it  is  prudent  to  state  unqualifiedly  that  reinforced  concrete 
beams  carrying  moving  loads  tending  to  produce  vibrations 
or  shock  should  be  so  designed  as  to  provide  for  the  entire 
transverse  shear  independently  of  the  shearing  resistance  of 


h     J 


C'       j' 


FIG.  3. 

the  concrete.  This  provision  for  resistance  to  transverse 
shear  is  made  chiefly  by  bending  upward,  in  that  part  of 
the  spans  near  the  end  supports,  the  steel  tension  reinforce- 
ment as  shown  in  Figs.  (2)  and  (3).  The  inclination  of 
the  bent  parts  of  the  rods  will  depend  upon  the  judgment 
of  the  engineer  in  view  of  the  length  of  the  span,  depth  of 
beam  or  other  features  of  each  case.  Usually  all  of  the 
rods  constituting  the  tension  reinforcement  are  not  bent 
upward,  as  that  much  provision  for  shear  is  not  needed.  If 
the  span  is  short,  there  may  be  but  one  set  of  bent  rods, 
as  shown  in  Fig.  2,  but  in  other  cases  there  may  be  two  or 
more  sets  bent  upward  at  different  distances  from  each 
end  of  the  span,  as  shown  in  Fig.  3,  the  number  of  such 
sets  of  rods  being  determined,  like  the  angle  of  inclination, 
in  accordance  with  the  best  judgment  of  the  designing 
engineer.  The  vertical  components  of  the  stresses  in  these 


624  CONCRETE-STEEL  MEMBERS.  [Ch.  XIII. 

bent  rods  may  obviously  equal  the  transverse  shear  at  the 
section  considered.  For  example,  in^Fig.  3,  the  total  trans- 
verse shear  at  the  section  CD  multiplied  by  the  tangent  of 
the  inclination  of  the  rods  to  the  vertical  must  for  good 
design  be  at  least  equal  to  the  required  horizontal  rein- 
forcement to  be  afforded  by  those  rods,  i.e.,  the  section  of 
the  rods  must  be  sufficient  to  take  their  stresses  without 
exceeding  the  prescribed  working  stress,  which  is  frequently 
16,000  pounds  per  square  inch.  A  similar  computation  is 
to  be  made  at  other  points  where  rods  are  to  be  bent  up- 
ward. It  is  not  necessary  (although  usual)  that  the  differ- 
ent sets  of  inclined  parts  of  rods  should  be  parallel,  i.e., 
those  nearer  the  centre  of  the  span  may  have  a  greater 
inclination  to  a  vertical  than  those  near  the  points  of 
support. 

Again,  vertical  reinforcing  pieces  or  stirrups,  such  as 
those  at  FH,  F'H',  CD,  CD',  in  Fig.  3,  are  introduced 
under  the  assumption  that  they  will  take  the  vertical  trans- 
verse shear  in  tension.  These  stirrups  are  of  a  variety  of 
forms  and  may  be  in  sets  of  two  or  more  vertical  prongs 
or  parts,  but  they  should  be  securely  fastened  to  the  hori- 
zontal steel  reinforcement.  If  the  total  transverse  shear  at 
any  section  as  JK  is  supposed  to  be  carried  by  the  stirrup 
as  tension  in  that  section,  the  cross-sectional  area  of  the 
steel  stirrups  should  be  sufficient  for  that  purpose  at  the 
prescribed  working  stress.  Furthermore,  the  adhesive  shear 
or  bond  on  the  exterior  surfaces  of  these  stirrups  should 
be  sufficient  to  give  such  tension  without  exceeding  the 
prescribed  working  stress  for  that  shear  or  bond.  These 
vertical  stirrups  are  thus  supposed  to  act  the  part  essen- 
tially of  vertical  truss  members  in  tension  and  so  produce 
diagonal  stresses  of  compression  in  the  concrete  as  shown 
by  broken  lines  in  the  vicinity  of  KC  and  K'C' .  It  is 
known  that  the  greatest  diagonal  stresses  of  tension  and 


Art.  ioo.]  SHEARING  STRESSES  AND  WEB  REINFORCEMENTS.       625 

compression  exist  at  angles  of  45  degrees  with  the  neutral 
surface  of  every  bent  beam.  The  function  of  these  stirrups 
is  intended  to  be  such  as  to  relieve  the  concrete  of  that 
tension  and  induce  diagonal  stresses  of  compression.  Indeed 
their  function  is  somewhat  similar  to  that  of  vertical  stiffen- 
ing members  on  the  web  plates  of  plate  girders  when  those 
stiffeners  are  assumed  to  take  tension  and  produce  com- 
pression in  the  web  in  a  4 5 -degree  direction,  as  was  fully 
shown  in  Art.  34.  The  distance  apart  of  these  vertical 
stirrups  should  certainly  not  be  greater  than  the  depth  of 
the  beam  from  the  upper  surface  down  to  the  tension  steel 
reinforcement;  probably  a  horizontal  distance  apart  of 
about  three-quarters  of  that  depth  is  advisable. 

If  any  beam  carry  a  set  of  loads,  W\,  W%,  Ws,  etc.,  and 
if  R'  is  the  end  shear  at  A,  Fig.  3,  and  if  IW  be  the  sum  of 
the  loads  between  the  end  A  and  any  section  at  which  it 
is  desired  to  obtain  the  transverse  shear  S',  then  will 
that  transverse  shear  at  any  stirrup,  as  CD,  Fig.  3,  be 
S'=R'—2W,  and  it  is  assumed  that  the  stirrup  will  carry 
that  shear  as  tension.  If  t'  is  the  allowed  tensile  stress  in 

S' 
the  stirrup,  the  sectional  area  As  of  the  latter  will  be  As  =— . 

If  the  intensity  of  permitted  bond  shear  is  sf  and  if  the  cir- 
cumference of  a  stirrup  section  is  o,  and  if  I'  is  the  imbedded 
length  of  one  complete  stirrup,  including  all  prongs,  then 
must  s'ol'  be  at  least  equal  to  S'.  Evidently  a  form  of 
cross-section  like  an  oblong  rectangle  will  give  much  more 
area  for  bond  shear,  for  a  given  sectional  area,  than  such  a 
section  as  a  circle  or  a  square  and  it  will  have  a  correspond- 
ing advantage  for  this  purpose. 

The  ends  of  all  stirrup  bars  as  well  as  all  reinforcing  rods 
should  be  turned  or  bent  at  right  angles  so  as  to  prevent 
slipping  at  and  near  .the  ends.  Furthermore,  they  should 
preferably  be  looped  at  top  and  bottom,  around  the  rein- 


626 


CONCRETE-STEEL  MEMBERS. 


[Ch.  XIII, 


forcing  rods  where  they  exist,  so  as  to  bear  directly  on  the 
concrete  supplementary  to  the  bond  shear. 

Obviously  if  both  the  inclined  bent  rods  shown  in  Figs. 
2  and  3  and  the  vertical  stirrups  shown  in  Fig.  3  effectively 
perform  their  functions,  both  would  not  be  needed  at  the 
same  part  of  a  beam,  but  as  the  effectiveness  of  each  detail 
by  itself  is  somewhat  uncertain,  both  are  frequently  used 
concurrently.  The  stirrups  may  judiciously  be  used  in  the 
central  part  of  the  span  extending  well  toward  the  ends 
where  the  bent  rods  are  employed. 

Another  form  of  steel  reinforcement  of  beams  is  shown  in 
Fig.  4,  which  is  supposed  to  be  part  of  a  reinforced  con- 


crete beam  on  both  sides  of  CD,  the  centre  of  span.  The 
beam  may  be  either  a  T-beam  or  a  beam  of  rectangular 
section.  The  steel  reinforcement  A B  is  supposed  to  be  on 
the  tension  side  of  the  beam  only,  although  a  precisely 
similar  reinforcement  might  be  placed  on  the  compression 
side  also.  The  small  inclined  bars  ab,  cd,  a'b',  c'd' ,  etc., 
are  usually  parts  of  the  main  tension  reinforcement  bent 
upward  in  a  diagonal  direction,  as  shown,  which  may  be 
at  the  angle  of  45  degrees  of  theory  or  at  some  other  angle. 
They  should  extend  above  the  neutral  surface  NS  and  be 
carried  nearly  to  the  top  of  the  beam. 

As  has  already  been  indicated,  a  solid  beam  of  a  single 
material  will  have  the  greatest  intensity  of  tensile  stress  at 


Art.  ioo.]  SHEARING  STRESSES  AND  WEB  REINFORCEMENTS.       627 

the  neutral  surface,  making  an  angle  of  45  degrees  with  a 
horizontal  line  and  sloping  upward  and  away  from  the 
centre  of  span,  as  indicated  in  Fig.  4.  Tests  of  plain  and 
reinforced  concrete  beams  show  that  in  those  parts  of  the 
span  near  the  end,  this  diagonal  tension  is  likely  to  cause 
failure  of  the  concrete.  Hence  these  inclined  bars  are  run 
up  from  the  main  tension  steel  reinforcement  to  assist  the 
concrete  in  taking  up  this  diagonal  tension  and  thus  pre- 
venting its  failure  so  far  as  possible.  The 'concrete  will  be 
in  compression  in  the  diagonal  direction  at  right  angles  to 
these  inclined  bars. 

If  a  vertical  section  of  a  beam  should  cut  two  or  more 
sets  of  them,  the  force  or  stress  obtained  by  multiplying 
half  the  total  transverse  shear  at  such  a  section  by  the 
secant  of  the  inclination  of  these  bars  to  a  vertical  line  will 
give  the  total  stress  to  which  those  two  or  more  sets  will 
be  subjected,  the  distribution  being  assumed  to  be  uniform 
among  them.  The  other  half  of  the  transverse  shear  at  that 
section  may  be  considered  as  giving  compression  to  the 
concrete  at  right  angles  to  the  4  5 -degree  tension  in  the 
inclined  bars.  It  is  clear  also  that  the  total  bond  shear 
on  the  surface  of  each  one  of  the  inclined  bars  must  be  at 
least  equal  to  the  tensile  stress  which  the  bar  carries  at  an 
intensity  not  greater  than  that  prescribed  in  the  speci- 
fications for  the  work.  If  such  a  normal  or  vertical  sec- 
tion of  the  beam  cuts  but  one  set  of  these  inclined  bars, 
the  single  set  must  take  the  stress  due  to  half  the  total 
transverse  shear,  precisely  as  described  above  for  two  or 
more  sets.  The  ends  of  these  inclined  bars  should  be  bent 
at  right  angles  or  otherwise  formed  so  as  to  prevent  the 
possibility  of  slipping,  and  thus  supplement  effectively  the 
bond  shear. 

It  is  clear  that  such  bars  must  add  to  the  carrying 
capacity  of  a  beam,  not  only  by  taking  up  the  inclined 


628  CONCRETE-STEEL  MEMBERS.  [Ch.  XIII 

tensile  stresses  as  described,  but  also  as  tending  to  bind  the 
entire  beam  together  as  a  unit. 

The  greatest  transverse  shear  is  that  at  the  ends  of 
the  span  where  the  sum  of  the  vertical  components  of  the 
stresses  in  the  bent  rods  must  be  equal  to  that  end  shear  or 
to  so  much  of  it  as  may  be  prescribed.  In  Fig.  3,  for 
example,  the  sum  of  the  vertical  components  of  the  stresses 
in  the  inclined  rods  bK  (a  set  of  reinforcing  rods)  must  be 
equal  to  the  transverse  shear  prescribed.  All  inclined  rods 
like  bK,  JD,  etc.,  lie  in  the  direction  of  the  diagonal  tension 
(maximum  at  inclination  of  45  degrees)  and  act  directly 
to  carry  shear. 

When  beams  are  continuous  over  supports,  as  shown  in 
Figs.  2  and  3,  bending  moments  will  be  developed  over  those 
supports  opposite  in  sign  to  those  found  at  and  near  the 
centres  of  the  spans,  producing  tension  in  the  upper  parts 
of  the  beams.  For  this  reason  tensile  steel  reinforcement 
formed  either  of  the  bent  rods  continued  into  the  adjoining 
spans,  as  shown  in  Fig.  3,  or  of  separate  rods  introduced 
for  the  purpose  are  required  to  take  that  tension. 

The  precise  degree  of  constraint  when  beams  or  girders 
are  "  continuous  "  over  points  of  support  cannot  be  deter- 
mined, but  certain  values  of  moments  expressing  the  results 
of  experience  in  modifications  of  formulae  for  conditions  of 
perfect  continuity  will  be  given  in  the  next  article. 

In  the  practical  consideration  of  provision  for  transverse 
shear  in  reinforced  concrete  beams,  it  is  a  matter  of  some 
uncertainty  how  much  the  concrete  may  be  allowed  to  take, 
if  any,  and  hence  what  corresponding  steel  must  be  intro- 
duced in  the  form  of  bent  reinforcing  rods  or  stirrups.  As 
has  been  intimated,  it  is  a  serious  question  whether  the 
concrete  should  be  credited  with  any  resistance  to  shear.  It 
is  frequently  the  practice  to  assume  that  one-third  of  the 
transverse  shear  will  be  carried  by  the  concrete  under  suit- 


Art.  ioi.]       STRESSES  IN  REINFORCED   CONCRETE  DESIGN.  629 

able  conditions  and  a  prescribed  working  stress,  but  that 
the  other  two-thirds  shall  be  taken  by  steel  provided  for 
the  purpose  as  already  described.  Aside  from  the  difficulty 
arising  in  the  attempt  to  distribute  by  measure  the  dis- 
charge of  an  important  function  between  two  different 
methods,  there  is  grave  doubt  about  the  propriety  of 
assuming  dependable  resistance  against  shear  by  con- 
crete, particularly  if  the  moving  load  is  of  a  character  to 
produce  vibrations  or  shock.  In  the  latter  case  steel  should 
certainly  be  provided  to  take  all  shear.  That  procedure 
is  more  prudent  in  all  cases  except,  possibly,  where  the  load 
is  wholly  dead  or  essentially  so,  when  the  concrete  may  be 
allowed  to  carry  one-third  of  the  total  transverse  shear. 

Many  tests  of  full-size  reinforced  concrete  T-beams  and 
beams  of  rectangular  section  have  been  made  by  Profs. 
Talbot,  Withey,  Hatt,  and  others  in  the  United  States  and 
by  Considere,  Feret  and  other  foreign  investigators  in 
Europe,  and  full  descriptions  of  all  results  may  be  found  in 
the  Bulletins  of  the  Universities  of  Illinois  and  Wisconsin 
and  in  many  other  publications,  hence  it  would  be  super- 
fluous to  repeat  them  here.  The  working  results  of  those 
tests  bearing  upon  computations  for  design  or  other  prac- 
tical work  will  be  given  in  the  next  article,  chiefly  in  con- 
nection with  the  recommendations  of  the  "  Report  of  the 
Committee  on  Concrete  and  Reinforced  Concrete  "  of  the 
American  Society  for  Testing  Materials,  Vol.  XIII,  1913. 

Art.  ioi. — Working  Stresses  and  Other  Conditions  in  Reinforced 
Concrete  Design.* — Design  of  T-beams. 

In  the  design  of  reinforced  concrete  beams  there  are 
some  features  of  the  work  determined  by  experience  and 

*The  report  of  the  Committee  on  Concrete  and  Reinforced  Concrete, 
Proc.  of  Am.  Soc.  for  Testing  Materials,  Vol.  XIII,  1913,  has  largely  been 
used  in  the  preparation  of  this  article. 


630  CONCRETE-STEEL  MEMBERS.  [Ch.  XIII. 

quite  independent  of  analysis.  Reinforcing  bars  or  rods 
must  be  surrounded  by  enough  concrete  to  receive  the 
proper  stress  from  the  -latter.  This  may  be  assumed  to  be 
done  if  the  lateral  spacing  between  the  centres  of  parallel 
rods  is  not  less  than  three  diameters,  or  two  diameters  from 
the  outer  concrete  surface  to  the  centre  of  the  nearest  rod, 
the  clear  vertical  space  between  two  horizontal  layers  of 
rods  being  not  less  than  i  inch.  It  is  seldom  advisable 
to  use  more  than  two  courses  of  such  rods.  In  all  cases 
scrupulous  and  effective  care  should  be  taken  by  the  aid 
of  blocking,  ties  and  other  devices  to  hold  the  reinforcing 
steel  accurately  in  place  until  the  concrete  is  set. 

As  a  fire  protection  a  thickness  of  at  least  2  inches 
of  concrete  should  be  placed  outside  of  the  steel  in  all  rein- 
forced concrete  beams  and  columns.  In  relatively  small 
beams  a  least  thickness  of  i^  inches  may  be  allowed,  and 
i  inch  may  be  permitted  in  floor  slabs. 

Floor  slabs  should  be  designed  and  reinforced  as  con- 
tinuous over  supports,  and  if  the  length  in  any  case  exceeds 
1.5  the  width  transverse  reinforcement  should  be  provided 
to  carry  the  entire  load. 

The  continuity  of  beams  and  slabs  may  be  recognized 
and  expressed  as  follows,  assuming  the  combined  dead  and 
moving  loads  equivalent  to  a  uniform  load  of  q  (pounds) 
per  linear  foot  on  the  effective  span : 

Floor  slabs:   moment  at  centre  of  span  and  at 

0/2 

end  of  span — . 

12 

Beams:  For  exterior  span  of  series,  moment  at 

al2 

centre  of  span  and  at  outer  fixed  end  of  span.  — . 

10 

For  interior  spans  moment   at   centre 

al2 

and  at  end  of  span — . 

12 


Art.  ioi.]     STRESSES  IN  REINFORCED   CONCRETE  DESIGN. 

Beams  and  Slabs:  continuous  over  two  spans 
only,  moment  at  central  support 

Moment  near  middle  of  span 


ql2 

-^— . 

8 

q? 

10 


At  ends  of  continuous  beams  and  girders  where  the 
degree  of  constraint  is  uncertain,  the  computation  of  the 
negative  end  bending  moment  must  be  controlled  by 
the  judgment  of  the  responsible  engineer. 

Working  Stresses. 

The  following  working  stresses  are  chiefly  given  as  per 
cents,  of  the  accompanying  ultimate  compressive  resistances. 
They  are  for  moving  and  dead  loads  considered  as  static 
loads,  with  the  assumption  that  proper  additions  to  moving 
loads  must  be  made,  when  advisable,  to  provide  for  impact 
or  vibrations. 

ULTIMATE   COMPRESSIVE   RESISTANCES 


Proportions  and  Ult.  Resistances,  Pounds  per 
Sq.  In. 


Aggregate. 

1:1:2 

i:ii:3 

1:2:4 

I    I  2j   :  5 

i  :3'6 

Granite,  trap  rock  
Gravel,      hard     limestone, 
sandstone  
Soft  limestone  and  sandstone 
Cinders 

3.300 
3,000 

2,200 
800 

2,800 

2,500 
i,  800 
700 

2,200 

2,000 
1,500 
600 

1,  800 

1,  600 
1,200 

soo 

1,400 

1,300 
1,000 

4.OO 

Working  Compression  in  Extreme  Fibre  of  Beam. 

The  working  intensity  in  the  extreme  compression  fibre 
of  a  beam  may  be  taken  at  32.5  per  cent,  of  the  ultimate 
compressive  resistance  as  determined  by  testing  concrete 
cylinders  8  inches  in  diameter  and  16  inches  high  at  the  age 


632  CONCRETE-STEEL  MEMBERS.  [Ch.  XIII. 

of  28  days.  If,  for  instance,  the  ultimate  compressive  resist- 
ance of  a  i  :  2  :  4  concrete  is  2200  pounds  per  square  inch, 
then  the  extreme  fibre  working  stress  would  be  2200X^25 
=  715  pounds  per  square  inch.  Adjacent  to  the  support  of 
continuous  beams,  these  stresses  may  be  increased  15  per 
cent. 

Shear  and  Diagonal  Tension. 

For  beams  with  horizontal  reinforcing  bars  only,  i.e., 
with  no  web  reinforcement,  2  per  cent,  of  the  ultimate  com- 
pressive resistance  may  be  allowed.  If  the  latter  were  2200 
pounds  per  square  inch,  as  for  the  1:2:4  concrete  of  the 
above  table,  the  allowed  shear  would  be  .02X2200=44 
pounds  square  inch.  This  shear  would  be  taken  wholly  by 
the  concrete. 

For  beams  thoroughly  reinforced  in  the  web,  6  per  cent, 
of  the  ultimate  compressive  resistance  may  be  allowed.  In 
this  case,  however,  the  web  reinforcement,  exclusive  of  bent- 
up  reinforcing  bars,  must  be  designed  to  take  two-thirds 
of.  the  external  vertical  shear.  Again,  using  the  1:2:4 
concrete,  the  allowed  shear  would  be  0.06X2200  =  132 
pounds  per  square  inch  of  total  concrete  section.  In  this 
case,  however,  the  steel  reinforcement  would  be  designed 
to  carry  two-thirds  of  the  total  transverse  shear,  making 
the  actual  shear  in  the  concrete  44  pounds  per  square  inch 
on  the  basis  of  the  exact  division  between  the  two  methods 
of  carrying  the  shear  prescribed. 

"  For  beams  in  which  part  of  the  longitudinal  reinforce- 
ment is  used  in  the  form  of  bent-up  bars  distributed  over  a 
portion  of  the  beam  in  a  way  covering  the  requirements  for 
this  type  of  web  reinforcement,  the  limit  of  maximum  ver- 
tical shearing  stress  "  may  be  taken  3  per  cent,  of  the  ulti- 
mate compressive  resistance. 

Where  what  is  termed  "  punching  shear  "  occurs,  i.e., 


Art.  ioi.]      STRESSES  IN  REINFORCED  CONCRETE  DESIGN.  633 

pure  shear  without  bending,  a  working  shearing  stress  of 
6  per  cent,  of  the  ultimate  compressive  resistance  may  be 
allowed. 

Bond  or  Adhesive  Shear. 

The  working  intensity  for  this  bond  or  shear  between 
concrete  and  plain  reinforcing  rods  may  be  taken  at  4  per 
cent,  of  the  compressive  resistance,  but  2  per  cent,  only  for 
drawn  wire.  For  i  :  2^  :  5  concrete  at  1600  pounds  per 
square  inch  of  ultimate  compressive  resistance,  the  two 
working  stresses  would  be  .04  X 1600  =64  pounds  per  square 
inch  or  half  that  for  drawn  wire. 

Steel  Reinforcement. 

The  tensile  or  compressive  working  stress  in  steel  rein- 
forcement should  not  exceed  16,000  pounds  per  square  inch. 

Modulus  of  Elasticity. 

For  computations  locating  the  neutral  axis  and  for  com- 
puting the  resisting  moment  of  beams  and  for  compression 
of  concrete  in  columns,  it  is  recommended  that  the  ratio 
of  the  steel  modulus  divided  by  the  concrete  modulus  be 
taken  at  15  if  the  ultimate  compressive  resistance  of  the 
concrete  is  taken  at  2200  pounds  per  square  inch  or  less; 
and  at  12  if  the  ultimate  compressive  resistance  of  the  con- 
crete is  greater  than  2200  pounds  per  square  inch  and  less 
than  2900  pounds  per  square  inch;  and,  finally,  at  10  if  the 
ultimate  compr.essive  resistance  of  the  concrete  is  taken 
greater  than  2900  pounds  per  square  inch. 

The  preceding  specifications  express  substantially  the 
views  of  a  Committee  on  Concrete  and  Reinforced  Concrete 
of  the  American  Society  for  Testing  Materials,  1913.  In 
that  Report  the  transverse  shear  is  computed  as  ifuni- 


634  CONCRETE-STEEL  MEMBERS.  [Ch.  XIII. 

formly  distributed  throughout  the  normal  section  of  a  bent 
beam,  which,  as  has  already  been  indicated,  is  incorrect. 
On  the  whole,  however,  the  recommended  values  are  judi- 
cious and  may  be  commended  for  practical  use. 

Design  of  T-Beam  for  Heavy  Uniform  Load. 

The  given  data  are  as  follows:  Effective  length  of  span 
32  feet  (non-continuous);  moving  load  on  floor,  250  pounds 
per  square  foot.  Floor  slab  6  inches  thick.  Each  T-beam 
carries  10  feet  width  of  floor. 

The  steel  reinforcement  of  the  beam  is  on  the  tension 
side  only. 

As  the  floor  slab  will  be  reinforced  (at  right-angles  to 
the  beam)  its  weight  will  be  taken  at  155  pounds  per  cubic 
foot.  The  concrete  will  be  considered  a  1:2:4  mixture 
weighing  about  150  pounds  per  cubic  foot. 

The  floor  slab  being  6  inches  thick,  a  little  less  than 
four  times  its  thickness  will  be  assumed  as  effective  com- 
pression flange  area  on  each  side  of  the  stem  or  web  of  the 
beam.  Referring  to  Fig.  i  of  Art.  96,  the  following  dimen- 
sional data  will  be  assumed  for  trial  computations  : 

b'  =  60  inches;  /  =  6  inches;  6  =  15  inches.  ^1=29 
inches;  thickness  of  concrete  outside  of  steel,  2  inches. 
Trial  value  of  steel  ratio  or  per  cent.,  p  =  .8  per  cent.  =  .  008. 

The  working  stresses  are  : 

Compression  for  concrete:  k\  =      650  Ibs.  per  sq.in. 
Tension  for  steel  :  /  =  16,000  Ibs.  per  sq.in. 


Eq.  (1.3)  of  Art.  96  then  gives: 

r^=  —i.  ii  ±1.524  =  +.414.-.  Ji  =  .414/11  =  12  ins. 


Art.  ioi.]  DESIGN  OF  T-BEAM.  635 

Eq.  (18)  of  Art.  96  may  now  be  used: 

p  = 5 X  .414  =  .0084  =  .84  per  cent. 

2  X  16,000 

This  last  value  of  p  agrees  closely  enough  with  the 
assumed  value.  Hence  the  computed  values  of  d\  and  p 
may  be  accepted.  Consequently:  ^3  =  29  —  12=17  inches. 

The  required  steel  sectional  area  is : 

A2=pb'hi  =.0084X60X29  =  14.62  square  inches. 

There  may  then  be  taken  eight  ij-inch  round  bars 
whose  aggregate  area  is  14.16  square  inches. 

The  cross-section  of  the  effective  beam  may  be  made 
as  shown  in  Fig.  i.  Deformed  rods  of  any  suitable  section 
of  the  aggregate  computed  area  may  obviously  be  used. 

The  dead  load  or  own  weight  of  the  beam,  including 
10  feet  width  of  floor  slab,  may  be  taken  at  1225  pounds 
per  linear  foot  of  span.  The  uniformly  distributed  moving 
load  will  be  10X250  =  2500  pounds  per  linear  foot  of  span. 
The  bending  moment  produced  by  these  two  loads  will  be: 

M  =  (25oo  +  i225)3°     3°Xi2  =5,028,800  inch  pounds. 

o 

The  resisting  moment  of  the  beam  section  must  now 
be  computed  by  the  aid  of  eq.  (8),  Art.  97. 

b  bf  j       ,  A<i  j 

-j-=I-25;    £~  =  4;    ai-/=o;    —=.944;    $3  =  17',  di=i2 

'  and  0  =  15. 

Introducing  these  numerical  quantities  in  eq,  (8),  Art.  97 : 

M  =  5, 024,611  inch-pounds. 
This  result  is  substantially  equal  to  the  external  bending 


636 


CONCRETE-STEEL    MEMBERS. 


[Ch.  XIII. 


moment  found  above  and  the  tentative  design  may  be 
accepted  as  satisfactory. 

There  still  remains  to  be  considered  suitable  provision 
for  end  and  intermediate  shears  which  will  be  made  by 
bending  upward  the  proper  number  of  reinforcing  rods 
supplemented  by  stirrups. 

Fig.  i  shows  to  scale  about  12 J  feet  of  the  T-beam,  the 
effective  cross-section,  60  inches  wide,  being  shown  in  shaded 
outline.  The  dimensions  are  self-explanatory  in  connection 
with  the  computations  already  made.  NS  is  the  neutral 


FIG.  i. 


axis.  The  eight  i^-inch  round  rods  in  two  courses  with 
their  central  line  4  inches  from  the  bottom  surface  are  shown 
both  in  section  and  in  longitudinal  broken  lines.  This 
latter  dimension  allows  a  fire-protecting  shell  of  concrete 
2  inches  thick  and  i  inch  clear  vertical  distance  between 
the  two  layers  of  four  rods  each. 

The  combined  dead  and  moving  load  on  the  beam  has 
already  been  shown  to  be  3725  pounds  per  linear  foot, 
making  the  end  shear  3725X15=55,875  pounds.  If  bent 
rods  inclined  at  an  angle  of  45°  be  supposed  to  take 
this  whole  shear,  the  total  stress  in  those  rods  will  be 
55,875  Xsec.  45  degrees  =  79,007  pounds.  If  the  steel  be 
stressed  at  16,000  pounds  per  square  inch,  a  little  less  than 


Art.  ioi.]  DESIGN  OF  T-BEAM.  637 

5  square  inches  of  section  will  be  required.  Three  ij-inch 
rounds,  or  their  equivalent  sectional  area,  will  supply  the 
desired  section.  It  will  be  convenient  to  bend  the  upper 
set  of  four  rods  as  shown  in  Fig.  i ,  thus  reducing  tho  actual 
stress  in  the  inclined  parts  to  about  12,000  pounds  per 
square  inch,  the  reduced  unit  stress  not  being  objectionable. 
A  greater  vertical  depth  of  concrete  would  have  been  avail- 
able for  shear  if  the  lower  set  had  been  bent  upward, 
but  with  the  use  of  stirrups  this  is  not  important  and  the 
arrangement  shown  is  a  little  more  convenient  in  actual  con- 
struction. If  desired  the  lower  set  could  be  bent,  but  it 
would  be  necessary  to  slightly  rearrange  the  position  of  all 
the  rods  so  that  the  bent  parts  of  the  lower  set  may  pass 
the  upper  set,  all  of  which  is  quite  feasible.  The  hori- 
zontal ends  of  the  bent  rods  should  also  be  bent  at  right 
angles  so  as  to  secure  the  firmest  possible  hold  on  the  con- 
crete at  the  end  of  the  beam.  The  horizontal  ends  of  the 
bent  bars  are  about  12  inches  long,  making  the  lower  bend 
of  the  same  rods  about  3.25  feet  from  the  end  of  the  beam. 

Vertical  stirrups,  24  inches  apart,  will  be  placed  through- 
out the  central  part  of  the  beam  and  they  will  be  carried 
down  so  as  to  pass  under  the  lower  reinforcing  rods.  There 
will  be  four  prongs  to  each  stirrup,  looped  at  top  and  bottom. 
By  this  arrangement  of  the  stirrups  the  bond  shear  on  their 
surfaces  is  greatly  reinforced  by  the  vertical  bearing  on  the 
concrete  and  reinforcing  rods  at  the  bottom.  The  first 
stirrup,  as  shown,  will  be  placed  at  the  lower  bend  in  the 
upper  set  of  reinforcing  rods,  although  the  stress  in  it  is 
indeterminate,  as  the  inclined  rod  is  supposed  to  take  the 
total  shear. 

The  total  transverse  shear  in  the  second  stirrup,  5.25  feet 
from  the  end  of  the  beam,  will  be  computed  as  carrying  in 
tension  9.75X3725=26,320  pounds,  requiring  at  16,000 
pounds  per  square  inch,  2.25  square  inches.  Four  if -inch  X 


638  CONCRETE-STEEL   MEMBERS.  [Ch.  XIII. 

f -inch  flat  bars  will  give  the  required  area,  each  such  flat 
bar  constituting  one  member  or  prong  of  the  stirrup.  The 
shear  at  the  next  stirrup  point,  2  feet  farther  from  the  end 
of  the  span,  will  be  28,870  pounds,  and  four  iJ-inchXife- 
inch  stirrup  sections  will  give  a  little  more  than  needed,  and 
that  jection  of  bar  will  be  adopted.  Although  smaller  bars 
would  be. sufficient  for  the  remaining  sections,  the  i|-inch 
X  A-inch  bars  will  be  retained  for  the  remaining  stirrups. 

The  total  available  concrete  section  for  resisting  shear 
is  29  inches  X 15  inches  =435  square  inches  which,  under  the 
specifications  of  the  preceding  article,  may  be  taken  at  44 
pounds  per  square  inch,  making  a  total  shear  of  19,140 
pounds  to  be  provided  for  in  this  way  if  it  should  be  con- 
sidered permissible.  If  the  latter  procedure  were  followed 
it  would  leave  but  two-thirds  of  the  total  transverse  shear 
at  each  stirrup  section  to  be  resisted  by  the  steel  stirrups. 
In  the  case  of  such  a  heavy  beam,  however,  it  is  believed 
to  be  the  better  practice  to  take  care  of  all  the  shear  by 
steel  reinforcement. 

If  4  5 -degree  steel  reinforcements  attached  to  the  main 
reinforcing  rods  were  used,  the  length  of  such  inclined  bars 
would  be  about  27  Xsec.  45  degrees  =38  inches.  Inasmuch 
as  half  the  transverse  shear  at  any  section  may  be  assumed 
to  produce  4  5 -degree  compression  at  right  angles  to  such 
inclined  tension  bars,  the  latter  may  be  computed  as  being 
stressed  by  half  the  transverse  shear  multiplied  by  sec.  45 
degrees.  The  4 5 -degree  tension  bars  near  the  end  of  the 
span  under  such  an  assumption  would  take  about  28,000 
pounds  only  and  if  there  were  four  of  them,  each  i^  inch  X 
TQ  inch,  they  would  be  sufficient.  At  intermediate  posi- 
tions further  removed  from  the  ends,  a  correspondingly 
smaller  section  might  be  used.  The  bond  shear  at  the  sur- 
face of  such  inclined  bars  could  be  taken  at  a  working 
stress  of  88  pounds  per  square  inch  of  surface.  Such  in- 


Art.  ioi.]  DESIGN  OF  T-BEAM.  639 

clined  tension  bars  should  be  placed  not  more  than  about 
21  inches  apart  horizontally  in  order  to  secure  effective 
action.  Their  upper  ends  should  be  bent  at  right  angles  or 
looped  to  secure  a  firmer  hold  on  the  concrete. 

These  computations  illustrate  clearly  the  simple  pro- 
cedures required  in  the  design  of  a  reinforced  concrete 
T-beam.  If  the  beam  is  of  rectangular  section,  the  pro- 
cedures are  precisely  the  same,  as  the  actual  rectangular 
section  in  that  case  would  correspond  precisely  to  the  effect- 
ive shear  section  taken  for  the  T-beam. 

Design  of  Continuous  Floor  Slab  for  6  Feet    Span    between 

Steel  Beams. 

The  slab  is  assumed  to  carry  a  warehouse  load  of  175 
pounds  per  square  foot  in  addition  to  own  weight.  It 
will  also  be  assumed  to  be  continuous  over  the  steel  beams 
6  feet  apart  centres,  the  degree  of  continuity  being  that 
prescribed  in  Art.  100,  making  the  centre  and  end  bending 

moments  each  — ,  w  being  the  load  per  lineal  foot  of  span. 

A  trial  depth  of  slab  of  4  inches  will  be  assumed  and  the 
design  will  be  made  for  a  12 -inch  width  of  slab.  A  depth 
of  i  inch  of  concrete  will  be  taken  outside  of  the  steel 
reinforcement,  which  will  be  wholly  on  the  tension  side  of 
the  slab,  and  the  tensile  resistance  of  the  concrete  will  be 
neglected.  The  data  to  be  used  will  then  be: 

Span  =6  feet.     Moving  load  =  1 7 5  pounds  per  square  foot. 
Dead  l^ad      =    50  pounds  per  square  foot. 

Tension  in  steel,  t  =  16,000  pounds  per  square  inch. 
Compression  in  concrete,  k\  =500  pounds  per  square  inch. 

7^=2=15;  h  =4  inches;  ^1=2.75  inches;  6  =  12  inches, 
-tii 


640  CONCRETE-STEEL  MEMBERS.  [Ch.  XIII. 

The  external  bending  moment,  M  =  225X    X    Xi2=8ioo 

12 

inch-pounds.     The  section   to   be   designed   must    give    a 
resisting  moment  at  least  equal  to  8100  inch-pounds. 
Eq.  (8),  Art.  98,  gives  the  steel  ratio: 

p  =  .005  =  .  5  per  cent. 
Hence, 

Ai  =.005  X4Xi2  =  .24  square  inch. 

Eq.  (4),  Art.  98,  then  gives  the  position  of  the  neutral 
axis  : 

^  =  -.075  ±.394  =  +.3  19; 

and 

d\  =0.88  inch; 

ds  =hi  —di  =  1*87  inches. 

The  internal  resisting  moment  will  .  now  be  given 
eq.  (12),  Art.  98: 


.88  \ 


=g7oo  inch_pounds 


By  revising  the  design  the  excess  above  8100  inch-pounds 
may  be  reduced  if  desired,  but  the  difference  is  too  small 
to  be  material. 

Two  f-inch  square  bars,  placed  6  inches  apart,  having 
a  combined  area  of  .28  square  inch,  will  afford  satisfactory 
reinforcement,  remembering  that  they  must  be  carried 
from  1  1  inches  above  the  lower  surface  of  the  slab  at  the 
centre  of  span  to  that  distance  below  the  upper  surface 
at  the  ends  of  the  span. 

The  end  shear  of  3X225=675  pounds  is  provided  for 


Art.  102.]  REINFORCED  CONCRETE  COLUMNS.  641 

by  the  bending  up  of  the  reinforcing  rods,  especially  as 
the  concrete  section  is  4X12  =48  square  inches. 


Art.  102. — Reinforced  Concrete  Columns. 

Reinforced  concrete  columns  may  be  divided  into 
two  classes.  The  reinforcing  steel  in  one  of  these  classes 
is  a  wrapping  or  banding,  usually  as  a  spiral,  of  the  concrete 
by  coarse  wire  or  thin  flat  bars,  so  that  the  lateral  strains 
or  enlargement  due  to  axial  compression  will  be  prevented 
as  much  as  possible  with  the  intent  to  increase  correspond- 
ingly the  carrying  capacity  of  the  col- 
umn. It  is  customary  to  use  longi- 
tudinal steel  rods  spaced  equidistantly 
around  the  column  adjacent  to  and 
inside  of  the  spiral  banding,  as  shown 
in  Fig.  2,  the  former  being  strongly 
fastened  to  the  latter  by  clamps  or 
wires.  When  the  cylindrical  cage 
thus  formed  is  filled  with  concrete,  FlG  x 

usually  a  rich  mixture  such  as  i  :  2  :  4, 

and  encased  with  concrete  about    2   inches  thick,  the  com- 
plete column  is  formed. 

The  steel  reinforcement  in  the  other  class  of  columns 
is  a  load-carrying  member,  in  fact  a  steel  column  in  itself, 
filled  with  concrete  and  encased  with  the  same  exterior 
shell  of  concrete  as  in  the  banded  column,  as  shown  in 
Fig.  3.  In  the  latter  case  the  parts  of  the  steel  column 
reinforcement  form  the  banding  or  wrapping  around  the 
concrete.  The  shape  of  cross-section  of  column  for  either 
class  may  be  any  desired,  although  the  circular  section 
is  more  convenient  for  the  first  class. 


642  CONCRETE-STEEL  MEMBERS.  [Ch.  XIII. 

Lateral  Reinforcement  and  Shrinkage 

The  analytic  expression  for  the  gain  in  carrying 
capacity  arising  from  banding  is  easily  written.  Let  Fig. 
i  represent  a  band  one  unit  (inch)  in  length,  i.e.,  along  the 
axis  of  the  column,  its  interior  diameter  being  d.  When 
the  column  receives  load  its  diameter  d  tends  to  increase 
in  consequence  of  the  lateral  strains,  thus  pressing  against 
the  interior  of  the  band  and  causing  the  latter  to  stretch 
accordingly.  Let 

£2  =30,000,000  =  modulus  of  elasticity  of  the  steel; 
EI  =  2,000,000  =  modulus  of  elasticity  of  the  concrete; 
px=  uniform  intensity  of  pressure  between  the  ring 

or  band  and  concrete; 

pi  =  intensity  of  column  loading  on  a  normal  section; 
6* = area  of  section  of  band; 
A = stretch  of  steel  ring  due  to  internal  pressure  px. 

Hence : 

A  =^~^d.    .'.  New  circumference  =wdl  i  +  ?«~)  (T) 

The  new  diameter  will  be  d[ 


2tE2, 

If  r  is  the^ratio  between  the  direct  compressive  and  lateral 
strains  for  concrete,  the  new  diameter  of  the  banded  con- 
crete will  be : 


EI 

Equating  the  two  values  of  the  new  diameter,  if  ~Er=ey 

ti&r^-^^r  •'•  t>=tJ-^—\  •  (3) 


Art.  102.]  REINFORCED   CONCRETE  COLUMNS.  643 

Eq.  (3)  gives  the  value  of  the  intensity  of  pressure  be- 
tween the  banding  and  the  concrete.     If  ^r  =  15,  and  if  r  =-, 

J^^  i  ^ 

(4) 


If  there  is  no  change  in  diameter,  eq.  3  gives, 


With  the  above  value  of  r,  p'x=—  would  prevent  all 

4 

lateral  strain,  and  as  eq.  (4)  shows  that  px  has  real  value, 
it  is  clear  that  the  banding  appears  to  be  highly  effective. 
Concrete,  however,  shrinks  when  it  sets  in  air  with  a 
coefficient  of  shrinkage,  according  to  such  tests  as  have 
been  made,  of  .0002  to  .0005.  If  £1  =  2,000,000  and  if, 
for  example,  £1  =  600  pounds  per  square  inch,  then  by 
eqs.  (2)  and  (4),  if  t  =  \  inch, 

/6oo    43  X4\ 


\ 

7 


2,ooo,ooo\  5  5     /     23,400 

As  both  and are  greater  than  ,  these 

5000          2000  23,400 

computations  show  that  shrinkage  of  concrete  setting  in  air 
will  more  than  neutralize  the  advantage  supposed  to  be 
due  to  banding,  at  least  until  the  elastic  limit  of  the  con- 
crete, and  probably  the  yield  point,  is  exceeded.  This 
explains  why  banding  shows  no  advantage  in  full-size 
column  tests  until  the  yield  point  is  passed,  as  will  be  seen 
later. 


644  CONCRETE-STEEL  MEMBERS.  [Ch.  XIII. 

Longitudinal  Reinforcement 

In  considering  the  effect  of  longitudinal  steel  reinforce- 
ment, let 

A  =  total  available  sectional  area  of  the  column   (the 
outer  2 -inch  thickness  is  neglected  in  computa- 
tions for  carrying  capacity) ; 
A 2  =  sectional  area  of  steel; 
A i  =  sectional  area  of  concrete; 

p=  steel  ratio  -/} 
J\. 

p\  =  intensity  of  compression  in  concrete; 
c  =  in  tensity  of  compression  in  longitudinal  steel; 

A=Ai+A2      and      *=:§?. 

tL\ 

P  =  carrying  capacity  of  reinforced  column ; 
pi=  carrying    capacity    of    plain    concrete    column    of 
section  A. 

Hence : 

.        .     (7) 


Or 

jr=p(e-i)+i (8) 

Eq.  (8)  shows  the  gain  of  carrying  capacity  due  to  the 
longitudinal  reinforcing  steel.     The  fractional  gain  is: 


(9) 


Many  tests  of  full-size  columns  of  both  types  have  been 
made  by  Professors  Talbot,  Withey,  the  author,  and  others, 
the  results  of  which  fully  described  may  be  found  in  the 
bulletins  of  the  Universities  of  Illinois  and  Wisconsin, 


Art.  102.]  REINFORCED  CONCRETE  COLUMNS,  645 

and  those  of  the  author  in  the  "  Proceedings  of  the  Insti- 
tution of  Civil  Engineers  "  of  London. 

The  effect  of  a  proper  amount  of  spiral  or  other  band- 
ing, either  by  itself  or  in  connection  with  longitudinal 
steel  rods  firmly  secured  to  it,  or  of  a  self-supporting  load- 
carrying  steel  column,  is,  in  all  these  ty^pes,  to  support  the 
concrete  to  such  a  degree  as  to  develop  substantially  its 
ultimate  carrying  power  in  short  blocks,  for  such  lengths 
of  columns  as  have  been  tested. 

In  order  to  accomplish  this  result  i  per  cent,  of 
lateral  steel  reinforcement  in  spiral  shape  is  sufficient. 
Furthermore  it  is  preferable  to  use  longitudinal  steel  rod 
reinforcement  in  connection  with  the  lateral  spiral  rein- 
forcement, the  two  being  firmly  attached  to  each  other 
in  all  cases.  The  spiral  cage  firmly  secured  to  the  longi- 
tudinal rods  constitutes  practically  an  independent  load- 
carrying  steel  column,  particularly  when  filled  with  con- 
crete. A  properly  designed  reinforced  column  of  this  type 
may  have  its  ultimate  carrying  capacity  closely  represented 
by  eq.  (7),  in  which  pi  is  the  ultimate  compressive  resist- 
ance of  the  concrete  and  c  the  ultimate  compressive  resist- 
ance of  the  steel.  If  longitudinal  rods  are  used  without 
the  steel  banding,  they  cripple  or  buckle  under  compara- 
tively light  loads,  as  would  be  expected,  and  make  an 
unsatisfactory  column  in  combination  with  the  concrete 
of  reduced  carrying  power. 

As  has  already  been  shown  in  connection  with  eqs.  (i)  to 
(5)  the  shrinkage  of  the  concrete  in  setting  prevents  the 
banding  influence  of  the  steel  from  being  effective  until 
the  yield  point  of  the  concrete  has  been  passed,  and  the 
results  of  tests  have  confirmed  fully  the  indications  of 
analysis.  The  same  tests,  however,  have  shown  that  in 
properly  designed  columns  of  both  classes  the  concrete 
and  the  steel  act  together  effectively  except  in  the  case  of 


646 


CONCRETE-STEEL  MEMBERS. 


[Ch.  XIII. 


longitudinal  rods  without  spiral  or  other  banding.  This 
latter  type  of  column,  however,  is  too  indifferent  in  char- 
acter to  be  used  in  practice. 

Types  of  Columns. 

Figs.    2   and  3  .illustrate  the  two  types  or  classes  of 
reinforced  concrete  columns  generally  used.     Fig.  2  shows 


FIG.  2. 


FIG.  3- 


a  spiral  reinforcement  inside  of  which  there  are  a  suitable 
number  of  longitudinal  round  or  other  rods  which  must 


Art.  102.]  REINFORCED-CONCRETE   COLUMNS.  647 

be  firmly  secured  to  the  spiral  reinforcement.  The  size 
of  the  latter  may  vary  according  to  the  size  of  the  column 
from  J  inch  diameter  to  f  inch  or  more,  and  the  pitch  may 
vary  from  i  to  several  inches,  according  to  the  size  of  the 
column.  It  has  been  found,  as  already  indicated,  that  the 
amount  of  spiral  or  lateral  reinforcement  should  be  about 
i  per  cent.,  i.e.-,  the  volume  of  the  spiral  metal  should  be 
about  i  per  cent,  of  the  volume  of  the  column,  counting  the 
diameter  of  the  latter  as  the  diameter  of  the  cylinder 
formed  by  the  centre  line  of  the  spiral.  The  amount 
of  longitudinal  steel  rods  may  be  i^  to  2  or  3  per  cent, 
or  more;  although  it  has  been  found  generally  to  be  more 
economical  to  increase  the  richness  of  the  concrete  core 
and  use  less  longitudinal  steel  than  to  use  more  of  the  latter 
with  leaner  and  less  expensive  concrete.  The  exterior 
concrete  shell,  usually  about  2  inches  thick,  is  not  con- 
sidered as  an  available  or  load-carrying  part  of  the  com- 
plete column.  It  has  been  found  by  experiment  that  this 
exterior  shell  may  carry  from  40  to  50  per  cent,  as  much 
load  per  square  inch  as  the  concrete  core,  and  that  it  will 
not  crack  off  under  test  until  the  yield  point  of  the  steel 
has  been  reached,  but  it  is  quite  likely  to  be  at  least  par- 
tially destroyed  in  a  burning  building.  On  the  whole, 
therefore,  it  is  considered  better  practice,  and  it  is  certainly 
safer  to  consider  the  core  only  of  reinforced  columns,  i.e., 
only  that  part  within  the  exterior  enveloping  volume  of 
the  steel  as  load  carrying. 

Fig.  3  is  typical  of  the  class  of  columns  in  which  the 
steel  is  designedly  a  load-carrying  member.  The  figure 
shows  a  column  of  four  angles  latticed  in  the  usual  manner 
with  batten  plates  as  well  as  lattice  bars  on  all  four  sides, 
but  a  great  variety  of  forms  may  obviously  be  used  in  this 
type  of  column.  Many  full-size  tests  have  shown  that  the 
concrete  filling  of  this  type  of  column,  no  less  than  in  the 


648  CONCRETE-STEEL  MEMBERS.  [Ch.  XIII. 

other  type,  may  be  considered  as  carrying  load  up  to  its 
full  ultimate  short  block  capacity  before  failure  of  the 
column  for  all  lengths  used  in  actual  tests.  The  load 
carried  by  such  a  column  may  be  computed  by  adding  the 
carrying  capacity  of  the  concrete  filling  considered  as  a 
short  block  to  the  carrying  capacity  of  the  steel  column 
computed  as  such. 

Prof.  Withey  has  concisely  expressed  the  results  of 
the  tests  of  full-size  columns  of  both  these  types  in  the 
Bulletin  of  the  University  of  Wisconsin  as  follows : 

"  i.  A  small  amount,  0.5  to  i  per  cent.,  of  closely  spaced 
lateral  reinforcement,  such  as  the  spirals  used,  will  greatly 
increase  the  toughness  and  ultimate  strength  of  a  concrete 
column,  but  does  not  materially  affect  the  yield  point. 
More  than  i  per  cent,  of  lateral  reinforcement  does  not 
appear  to  be  necessary.  The  use  of  lateral  reinforcement 
alone  does  not  seem  advisable. 

."  2.  Vertical  steel  in  combination  with  such  lateral 
reinforcement  raises  the  yield  point  and  ultimate  strength 
of  the  column  and  increases  its  stiffness.  Columns  rein- 
forced with  vertical  steel  only  are  brittle,  and  fail  suddenly 
when  the  yield  point  of  the  steel  is  reached,  but  are  con- 
siderably stronger  than  plain  columns  made  from  the  same 
grade  of  concrete. 

..."  3.  Increasing  the  amount  of  cement  in  a  spirally 
reinforced  column  increases  the  strength  and  stiffness  of 
the  column.  A  column  made  of  rich  concrete  or  mortar 
and  containing  small  percentages  of  longitudinal  and 
lateral  reinforcement,  is  without  doubt  fully  as  stiff  and 
strong  and  more  economical  than  one  made  from  a  leaner 
mix  reinforced  with  considerably  more  steel.  In  these 
tests,  doubling  the  amount  of  cement  increased  the  yield 
point  and  ultimate  strength  of  the  columns  without  vertical 
steel  about  100  per  cent.,,  and  added  about  50  per  cent,  to 


Art.  102.]  REINFORCED    CONCRETE  COLUMNS.  649 

the  strength  of  those  reinforced  with  6.1  per  cent,  vertical 
steel. 

"4.  From  the  behavior  under  test  of  the  columns 
reinforced  with  spirals  and  vertical  steel  and  the  results 
computed,  it  would  seem  that  a  static  load  equal  to  from 
35  to  40  per  cent,  of  the  yield  point  would  be  a  safe  working 
load. 

"5.  The  results  obtained  from  tests  of  columns  rein- 
forced with  structural  steel  indicate  that  such  columns 
have  considerable  strength  and  toughness,  and  that  the 
steel  and  concrete  core  act  in  unison  up  to  the  yield  point 
of  the  former.  The  shell  concrete  will  remain  intact  until 
the  yield  point  of  the  steel  is  reached,  but  no  allowance 
should  be  made  for  its  strength  or  stiffness." 


"2.  Although  the  yield  point  of  a  reinforced  concrete 
column  is  practically  independent  of  the  percentage  of  spiral 
reinforcement,  the  ultimate  strength  and  the  toughness  are 
directly  affected  by  it.  ...  Consequently,  only  enough 
lateral  reinforcement  is  needed  to  prevent  the  longitudinal 
rods  from  bulging  outward,  and  to  provide  an  additional 
factor  of  safety  against  an  overload  by  increasing  the 
toughness  and  raising  the  ultimate  strength  somewhat 
above  the  yield  point.  From  these  tests  i  per  cent,  of  a 
closely  spaced  spiral  of  high-carbon  steel  seems  to  be 
sufficient  for  this  purpose. 

"3.  By  the  addition  of  longitudinal  steel  the  yield 
point,  ultimate  strength  and  stiffness  of  a  spirally  rein- 
forced column  can  be  considerably  increased.  If  maximum 
economy  in  floor  space  is  desired,  if  a  column  is  so  long  or 
is  so  eccentrically  loaded  that  tension  exists  on  a  portion 
of  the  cross-section,  or  if  a  large  dead  load  must  be  sus- 
tained by  the  column  while  the  concrete  is  green,  a  high 


650  CONCRETE-STEEL  MEMBERS.  [Ch.  XIII 

percentage  of  longutidinal  reinforcement  may  often  be 
advantageously  employed.  Such  reinforcement  is  also  a 
valuable  safeguard  against  failure  due  to  flaws  in  the 
concrete.  If  the  cost  of  cement  is  extremely  high,  it  may 
be  economical  to  use  a  leaner  mixture  than  suggested  in 
(i)  and  considerable  longitudinal  steel  to  increase  the 
stiffness  and  strength;  columns  like  those  of  Series  i 
may  profitably  be  used.  In  general,  however,  cement  is 
a  more  economical  reinforcement  than  steel.  Therefore, 
for  ordinary  constructions  it  does  not  seem  advantageous 
to  use  in  combination  with  a  rich  concrete  more  than  2 
or  3  per  cent,  of  longitudinal  steel." 

"8.  Briefly  summarizing  the  foregoing,  it  seems  eco- 
nomical to  use  for  reinforced  concrete  columns  a  very  rich 
mixture,  and  advantageous  to  employ  about  i  per  cent,  of 
closely  spaced  high-carbon  steel  lateral  reinforcement  com- 
bined with  2  or  3  per  cent  of  longitudinal  reinforcement. 
From  the  test  data -presented  it  seems  apparent  that  such 
columns,  centrally  loaded,  may  be  subjected  to  a  static 
working  stress  equal  to  one-third  of  the  stress  at  yield 
point." 

Working  Stresses 

The  results  of  analysis  and  of  the  full-size  tests  to 
which  reference  has  been  made  furnish  a  rational  basis 
on  which  proper  working  stresses  may  be  based.  The 
concrete  is  so  held  and  supported  in  both  types  of  column, 
when  properly  designed,  that  the  working  stress  in  it  may 
be  prescribed  as  if  it  were  a  short  block.  In  that  class  of 
columns  in  which  the  steel  reinforcement  is  a  steel  column 
by  itself,  the  working  stress  in  the  latter  may  be  prescribed 
precisely  as  for  any  other  steel  column.  Manifestly  the 
fraction  of  the  ultimate  resistance  represented  by  the  work- 
ing stresses  for  both  materials  must  be  the  same.  Actual 


Art.  102.]  REINFORCED  CONCRETE  COLUMNS.  651 

tests  of  full-size  columns  enable  the  unit  working  stress 
for  the  longitudinal  steel  in  the  spiral-banded  columns  to 
be  properly  prescribed,  the  steel  spiral  banding  being  a 

1  per  cent,   lateral  reinforcement  not  to  be   credited  as 
carrying  any  direct  load. 

The  unit  compressive  working  stress  of  the  longitudinal 
steel  reinforcing  members  in  either  type  of  column  is  taken, 
in  the  recommendations  of  the  American  Society  for  Testing 
Materials  in  their  Proceedings  for  1913,  at  16,000  pounds 
per  square  inch,  it  being  understood  that  the  length  of  no 
column  shall  exceed  fifteen  times  the  least  width,  that 
width  not  including  the  protective  shell,  usually  about 

2  inches  thick. 

The  same  ratio  of  length  to  least  width  holding  for 
both  types  of  columns,  the  following  compressive  working 
stresses  are  recommended  by  the  Committee  on  Concrete 
and  Reinforced  Concrete  of  the  American  Society  of  Civil 
Engineers,  1913,  the  per  cents,  stated  to  be  applied  to  the 
ultimate  resistances  of  the  various  grades  of  concrete  given 
in  Art.  101. 

Structural  steel  in  tension 16,000  Ibs  per  sq.in. 

Per  cent,  of 

Ult.  Com- 

press!  ve 

Resist. 

Concrete   in   compression   where   resisting   area   is   at   least   twice 

loaded  area 32 . 5 

Concrete  in  plain  concrete  column  or  pier  centrally  loaded,  length  = 

12  diameters  or  less 22 . 5 

Concrete  in  column  with  I  to  4  per  cent,  longitudinal  reinforcement 

only;    length  of  column  =  12  diameters  or  less 22 .5 

Concrete  in  column  with  lateral  reinforcement  of  spirals,  etc.,  at 
least  i  per  cent,  of  volume  of  column,  clear  spacing  of  spirals  or 
hooping,  ro  to  £  of  diameter 'of  encased  column,  in  no  case  ex- 
ceeding 2 1  inches,  the  length  of  laterally  unsupported  column  to 

be  not  more  than  8  diameters"  of  hooped  core 27 . 

Concrete  in  column  with  i  per  cent,  to  4  per  cent,  of  longitudinal 
bars  with  spirals,  hoops,  etc.,  as  specified  above  column,  the 
length  of  laterally  unsupported  hooped  core,  not  more  than  8 
diameters  of  core 32 . 625 


652  CONCRETE-STEEL  MEMBERS.  [Ch.  XIII. 

Reinforced  columns  with  longitudinal  steel  rods,  only, 
embedded  in  the  concrete  are  highly  unsatisfactory  and 
they  should  not  be  used  where  the  failure  of  the  column 
would  entail  serious  consequences. 

The  tests  of  such  load-carrying  columns  for  steel  rein- 
forcement of  reinforced  concrete  as  have  been  made  by 
the  author  show  that  their  ultimate  resistances  will  be  closely 
given  for  such  lengths  as  have  been  tested  by  the  simple 
straight-line  formula 

P  I 

-=43,000-155-. 


In  this  formula  -  is  the  ratio  of  length  of  column  to 

the  radius  of  gyration  of  its  cross-section  about  the  neutral 

p 

axis  and  A  is  the  area  of  cross-section.     Hence  —  is  the 

A 

average  unit  compressive  stress  over  a  normal  section  of 
column. 

This  type  of  column  is  not  limited  in  use  to  any  ratio 
of  length  over  least  diameter,  nor  is  the  per  cent,  of  steel 
section  restricted.  As  the  steel  reinforcement  is  a  perfectly 
designed  load-carrying  column,  it  may  be  treated  like  any 
other  steel  column,  while  the  concrete  filling  is  so  banded 
and  supported  by  the  enclosing  steel  column  that  load 
may  be  imposed  upon  it  as  in  the  case  of  a  short  concrete 
block. 

These  columns  have  been  used  for  tall  buildings  of 
eleven  stories  or  more  in  height.  They  are  well  adapted 
to  such  a  purpose,  not  only  in  consequence  of  the  load- 
carrying  capacity  of  the  steel,  but  also  on  account  of  the 
facility  with  which  floor  beams  and  girders  or  other  members 
may  be  detailed  to  them. 


Art.  102.]  PROBLEMS  FOR  CHAPTER    XIII.  653 

The  spiral  or  otherwise  banded  column  is  not  so  well 
adapted  to  structural  purposes.  The  design  is  such  that 
they  are  available  only  for  comparatively  short  lengths  in 
connection  with  the  prescribed  working  stresses.  They  may 
probably  be  used  up  to  lengths  of  unsupported  core  equal 
to  twelve  times  the  least  diameter  under  a  reduction  of 
working  stresses  to  80  per  cent,  of  those  prescribed. 

The  two  following  problems  will  illustrate  the  applica- 
tions of  the  preceding  results  to  actual  design  work: 

PROBLEM  I. 

Design  a  reinforced-concrete  column  13  feet  6  inches 
long,  with  spiral  banding  and  longitudinal  rod  reinforce- 
ment to  carry  a  load  of  354,000  pounds. 

As  the  column  must  not  exceed  8  diameters  in  length, 
the  diameter  of  the  spiral  banding  will  be  taken  as  20 
inches,  giving  an  effective  area  of  314.2  square  inches. 
There  will  be  assumed  eight  if -inch  longitudinal  round 
rods  arranged  as  shown  in  Fig.  2.  The  concrete  will  be 
taken  as  a  i  :  2  :  4  mixture  with  an  ultimate  resistance 
at  twenty-eight  days  of  2250  pounds  per  square  inch. 
Hence  the  working  unit  stress  will  be  1^  =  2250X32. 625  =734 
pounds  per  square  inch.  The  working  stress,  c,  of  the  steel, 
as  has  been  shown  by  the  specifications  of  the  joint  com- 
mittee of  the  Am.  Soc.  C.E.  and  the  Am.  Soc.  for  Testing 
Materials,  may  be  taken  at  16,000  pounds  per  square  inch. 
Hence  the  total  carrying  capacity  of  the  column  is: 

Of  the  steel  section 8  X  16,000  =128,000  Ibs. 

Of  the  concrete  section.  (314.2-8)  X734  =  244,751  Ibs. 

Total - 

352,751  IBs. 

This  is  sufficiently  near  354,000  to  be  considered  satis- 
factory and  it  will  be  accepted*  It  illustrates  fully  the 


654  CONCRETE-STEEL  MEMBERS.  [Ch.  XIII. 

procedure  to  be  followed  in  the  design  of  this  type  of 
column. 

The  i  per  cent  of  spiral  lateral  reinforcement  may  be 
determined  as  follows:  The  volume  of  spiral  metal  per 
inch  of  length  of  column  is  0.01X314=3.14  cubic  inches. 
If  the  pitch  is  2  inches  (one-tenth  the  diameter)  the  length 
of  one  complete  turn  of  the  spiral  will  be  about  63  inches. 

Hence  the  sectional  area  of  the  spiral  rod  will  be  — —  = .  i 

63 

square  inch  (nearly),  requiring  a  f-inch  round  rod.  This 
close  wrapping  or  banding  by  a  f-inch  spiral  with  2 -inch 
pitch  must  be  firmly  fastened  by  coarse  wire  or  clips  to 
the  eight  if -inch  longitudinal  round  rods. 

PROBLEM  II. 

Design  a  reinforced-concrete  column  20  feet  long  to 
carry  a  load  of  283,000  pounds,  the  steel  reinforcement  to 
be  a  load-carrying  column. 

Let  the  reinforcing  column  be  composed  of  four  3  X3  X re- 
inch  steel  angles  latticed  to  form  a  column  like  Fig.  3. 
The  square  formed  by  the  angles  will  be  15  inches  on  a 
side,  i.e.,  from  back  to  back  of  angles.  The  radius  of 

gyration  r  of  such  a  section  is  6. 7  inches.     Hence  -  =  -^—  =36, 

r     6.7 

p 
and  eq.  (10)  gives -r  =37,420  pounds  per  square  inch.     If 

A. 

working  stresses  be  taken  at  one-third  the  ultimate,  the 

working  stress  for  steel  will  be  g=3      '-  =  12,470  pounds 

o 

per  square  inch.  The  sectional  area  of  a  3X3  X^-inch 
angle  is  2.43  square  inches.  Hence  the  effective  area 
of  the  concrete  section  is  15X15—  4X2. 43  =215. 3  square 
inches.  The  concrete  will  be  assumed  to  be  a  1:2:4 
mixture,  for  which  the  ultimate  resistance  may  be  taken 


Art.  103.]  DIVISION  OF  LOADING.  655 

at  2250  pounds  per  square  inch,  and  the  working  resist- 
ance, 750  pounds  per  square  inch.  The  total  carrying 
capacity  of  the  column  will  then  be: 

Of  the  steel  section 12,470X9.72  =121,240  Ibs. 

Of  the  concrete  section .       750X215.3=16 1 ,460  Ibs. 


Total 282,700  Ibs. 

This  result  shows  that  the  design  is  satisfactory. 

Art.  103. — Division  of  Loading  Between  the  Concrete  and  Steel 
Under  the  Common  Theory  of  Flexure. 

It  is  occasionally  desirable  to  determine  the  portion  of 
the  total  loading  of  either  a  concrete-steel  beam  or  arch 
carried  by  the  steel  and  concrete  parts  of  the  member. 
In  making  this  determination  the  formula  established  in 
the  preceding  articles  in  accordance  with  the  common 
theory  of  flexure  will  be  employed.  It  will  be  convenient 
also  for  this  purpose  to  represent  the  intensity  of  stress  in 
the  extreme  fibre  of  the  steel,  whether  tension  or  com- 
pression, by  kv  the  distance  of  that  extreme  fibre  from  the 
neutral  axis  of  the  composite  section,  established  in  Art.  97, 
being  represented  by  d%.  It  will  further  be  supposed  that 
the  coefficients  of  elasticity  for  concrete  in  tension  and 
compression  are  the  same.  Eqs.  (4)  of  Arts.  96  and  98, 
representing  the  resisting  moment  of  the  internal  stresses 
in  a  normal  section  of  a  composite  member,  may  then  be 
written 

kill       &2/2  ,    x 

M=^r+^ (i) 

Let  the  total  load  on  the  composite  beam  or  arch  be 
represented  by  W,  while  W\  and  W2  represent  the  portions 


6S6  CONCRETE-STEEL  MEMBERS.  [Ch.  XIII. 

of  W  carried  by  the  steel  and  concrete  respectively.     Also 

W  W2 

let  q1  and  q2  be  so  taken  that  ^-^r  and  <?2  =  ^7.     The 

remaining  notation  will  be  that  given  in  Art.  96. 

Since  the  bending  moments  in  the  portions  Al  and  A2 
are  proportional  to  the  loads  which  those  portions  carry, 

k  k 

remembering  that  -r  and  -f  are  equal  to  E±u  and  E2u  re- 
Gfj  a2 

spectively,  there  may  be  written,  as  indicated  by  eq.  (i), 

_^_    E^       and   q-E>-  _IA 

and     q,-w- 


Also  K     (2) 

I, 

and     a.  =  - 


and 


W 
Also,  if  n==> 


Then,  since     Ml=qlM  and  M2  =qzM 

k^-f     and     *,=.       ...     (4) 

*1  ^2 

Eqs.  (2),  (3),  and  (4)  show  the  portions  of  loading 
carried  by  the  two  materials  and  the  greatest  intensities 
of  stresses  in  their  extreme  fibres. 

It  is  sometimes  necessary  to  combine  a  bending  moment 
with  the  direct  compression  (or  tension)  produced  by  a 
force  P  acting  along  or  parallel  to  the  axis  of  a  beam  or 
arch.  Let  pl  and  p2  represent  the  intensities  of  stress 
produced  in  the  two  portions  A1  and  A2  by  such  a  direct 
force.  Since  equal  unit  longitudinal  strains  exist  in  the 


Art..  103.]  DIVISION   OF  LOADING.  657 

two  materials,  the  intensities  of  stress  in  the  portions  Al 
and  A2  will  be  proportional  to  their  coefficients  of  elastici- 
ties. Hence 


Hence 

•     •     (6) 


In  the  case  of  an  elastic  arch  like  those  of  combined 
concrete  and  steel,  the  thrust  P  is  in  general  exerted  along 
the  axis  of  the  arch  ring  but  at  some  distance,  /,  from  it. 
In  such  a  case  the  bending  moment  is 

M=Pl',     hence     M^qfl     and     M2=q2Pl.     .     (7) 

The  values  of  the  bending  moments  are  to  be  placed  in 
eq.  (4),  in  order  to  determine  the  intensities  kl  and  k2. 

In  determining  the  resultant  of  stress  for  any  section 
of  an  arch  ring,  if  the  conditions  under  which  eqs.  (2)  were 
written  be  employed,  the  thrust  on  the  portion  Al  will  be 
qfy  and  qf  on  A,,  since  the  thrusts  on  the  two  portions 
will  be  proportional  to  the  loads  which  they  carry.  Hence, 
if  k^  and  k2  again  be  used  to  represent  the  greatest  inten- 
sities of  stress  in  the  two  portions,  there  may  at  once  be 
written 


P      Md\ 

±-r*) (9) 


In  eqs.  (8)  and  (9),  M=Pl. 

If,  again,  the  last  members  of  eqs.  (5)  and  (6)  be  used 


658  CONCRETE-STEEL  MEMBERS.  [Ch.  XIII. 

in   connection  with  eqs.   (2)  and  (4)  the   resultant  values 
of  &j  and  k2  will  be 

P 


P 


In  the  use  of  all  these  equations,  care  must  be  taken 
to  give  the  proper  sign  to  the  bending  moment  M. 

These  equations  comprise  all  that  are  necessary  in  order 
to  ascertain  the  distribution  of  the  loading  between  the 
steel  and  the  concrete,  or  any  other  two  materials,  whether 
the  case  may  be  one  of  pure  bending  or  a  combination  of 
bending  and  direct  stress. 


CHAPTER  XIV. 

ROLLED   AND   CAST  FLANGED   BEAMS 

Art.  104.—  Flanged  Beams  in  General. 

ROLLED  flanged  beams  as  produced  by  steel  mills  and 
used  in  building  or  other  construction  have  already  been 
treated  in  cases  of  simple  bending,  using  the  moment  of 
inertia  either  by  itself  or  as  part  of  the  section  modulus 
for  steel  beams  where  their  moments  of  resistance  take 
the  usual  form, 


(i) 


In  this  equation  d\  is  the  distance  of  the  extreme  fibre 
from  the  neutral  axis  in  which  the  intensity  of  stress  k 

exists,  and  I  and  —  are  the  moment  of  inertia  and  the 
01 

section  modulus,  respectively,  numerical  values  of  which 
for  all  shapes  are  given  in  handbooks.  In  this  treatment 
of  rolled  or  other  flanged  beams  the  resistance  of  the  web  is 
included,  but  there  are  cases  when  it  is  permissible  to  neglect 
the  bending  resistance  of  the  web  or,  again,  in  which  the 
bending  resistance  of  the  two  flanges  is  treated  separately, 
as  if  the  intensity  of  stress  in  each  is  uniform  throughout 
the  flange  section,  to  which  a  closely  approximate  simple 
expression  for  the  bending  resistance  of  the  web  may  or 
may  not  be  added. 

If  the  bending  resistance  of  the  flanges  is  to  be  com- 

659 


660  ROLLED  AMD  CAST  FLANGED  BEAMS.  [Ch.  XIV. 

puted  by  itself,  it  is  evident  that  economy  of  design  requires 
that  the  two  flanges  must  fail  concurrently  if  the  beam 
be  loaded  to  failure.  If  the  ultimate  tensile  and  com- 
pressive  resistances  of  the  material  are  not  the  same,  it 
is  equally  clear  that  the  two  flanges  should  not  be  of  equal 
section,  the  area  of  the  flange  in  which  the  ultimate  resist- 
ance is  greater  being  less  than  that  of  the  flange  in  which 
the  ultimate  resistance  is  less.  This  results  from  the 
fact  that  the  total  stress  of  compression  in  the  compres- 
sion flange  must  be  equal  to  the  total  tensile  stress  in  the 
tension  flange,  the  beam  being  supposed  to  be  horizontal 
and  the  load  vertical.  If  the  bending  resistance  of  the 
web  is  recognized,  the  equality  of  the  two  total  flange 
stresses  no  longer  holds,  since  the  tension  and  compression 
developed  in  the  web  is  to  be  added  to  the  corresponding 
stresses  in  the  flanges  in  order  to  make  equality. 

Each  total  flange  stress  is  evidently  equal  to  the  flange 
area  multiplied  by  the  intensity  of  assumed  uniform  stress 
in  it.  The  centre  of  each  flange  stress  will  then  be  the 
centre  of  gravity  of  the  section  on  which  it  acts.  The 
vertical  distance  d  between  the  centres  of  gravity  or  stress 
of  the  two  flanges  is  called  the  effective  depth  of  the  beam, 
because  if  it  be  multiplied  by  either  flange  stress  the  prod- 
uct will  be  the  resisting  moment  of  the  stresses  acting 
in  the  section  of  the  beam.  In  other  words  the  effective 
depth  d  is  the  lever  arm  of  the  internal  couple  whose  moment 
is  equal  to  the  external  bending  moment. 

Let  a  be  the  sectional  area  of  the  tension  flange  and  T 
the  uniform  intensity  of  stress  in  it,  and  let  a'  and  C  be 
the  corresponding  values  for  the  compression  flange,  while 
d  is  the  effective  depth.  Then,  since  aT  =  a'C,  the  moment 
of  the  internal  stresses  will  be 

M=aTd=a'Cd.  (2) 


Art.  105.]       FLANGED  BEAMS   WITH  UNEQUAL  FLANGES.  661 

The  use  of  both  eqs.  (i)  and  (2)  will  be  illustrated  by 
numerous  practical  applications. 

It  is  clear  from  what  has  preceded  that  the  chief 
function  of  the  flanges  is  to  resist  the  bending  proper, 
while  the  main  function  of  the  web  is  to  resist  the  trans- 
verse shear. 

The  direct  stresses  of  tension  and  compression  in  a 
beam  with  solid  rectangular  section  correspond  to,  i.e., 
perform  the  same  function  as,  the  flange  stresses  of  tension 
and  compression  in  the  flange  beam,  while  the  web,  supposed 
to  take  shear  only,  corresponds  approximately  to  the  zone 
of  material  in  the  vicinity  of  the  neutral  surface  of  the 
solid  section  in  which  the  direct  stresses  of  tension  and 
compression  are  either  zero  or  nearly  zero. 

Art.  105.— Flanged  Beams  with  Unequal  Flanges. 

By  the  common  theory  of  flexure,  if  the  two  coefficients 
of  elasticity  are  equal,  it  has  been  shown  that  if  C>  Fig.  i, 
is   the   centre   of   gravity   of    the          h~~~&"~      "71  _ 
cross-section,     the     neutral     axis          I 
of  the   section  will   pass  through 
that  point.     Let  it  now  be   sup- 
posed that  the  lower  flange  is  in      ^__ 
tension,  while  the  upper  is  in  com- 
pression.    Also    let     T    represent      I       ^ 
the  ultimate  resistance  to  tension    Di 


in  bending,  and  let  C  represent  the 
same  quantity  for  compression  in  FIG.  i. 

bending.     Then   s:nce  intensities  vary  directly  as  distances 
from  the  neutral  axis, 

' 


662  ROLLED  AND  CAST  FLANGED  BEAMS.  [Ch.  XIV. 

The  ratio  between  ultimate  intensities  is  represented  by 
nf.     If  d=k  +  hi  is  the  total  depth  of  the  beam,  and  hence 


ld 


. 


If,  as  an  example,  for  cast  iron  there  be  taken 

"T^ 

n'=—  =0.2,     hi=-d. 
L  o 

The  relation  between  h  and  h^  shown  in  eq.  (2)  is  en- 
tirely independent  of  the  form  of  cross-section.  But 
according  to  the  principles  just  given,  in  order  that  economy 
of  material  shall  obtain,  the  cross-section  should  be  so  de- 
signed that  h  and  h^  shall  represent  the  distances  of  the  centre 
of  gravity  from  the  exterior  fibres. 

The  analytical  expression  for  the  distance  of  the  centre 
of  gravity  from  DF  is 

£frV+  (b-V)f(d-tf)  +K61~fr')*12  ,  x 

-' 


(6  -&'X  +  (&i-&'K 

The  meaning  of  the  letters  used  is  fully  shown  in  the 
figure.  In  order  that  the  beam  shall  be  equally  strong  in 
the  two  flanges,  the  various  dimensions  of  the  beam  must 
be  so  designed  that 

*!=/*!  .......         •          (4) 

It  would  probably  be  found  far  more  convenient  to  cut 
sections  out  of  stiff  manila  paper  and  balance  them  upon 
a  knife-edge. 


Art.   105.]       FLANGED  BEAMS   WITH  UNEQUAL  FLANGES.  663 

The  moment  of  inertia  about  the  axis  AB,  thus  deter- 
mined, is 


.    (4a) 

.  to  be  substituted  in  the  formula  71 
now  changed  to 


kl 
This  value  is  to  be  substituted  in  the  formula  M=— , 


<•» 


For  various  beams  whose  lengths  are  /  and  total  load  W 
the  greatest  value  of  M  becomes  : 
Cantileve    uniformly  loaded, 

Wl 
M=  —  . 

2 

Can'ilever  loaded  at  end, 


Beam  supported  a'  each  end  and  uniformly  loaded  t 
M-^-£ 

~     Q      "  ~    Q     ' 
O  O 

Beam  supported  a  each  end  and  loaded  at  centre, 

Wl 

M=  —  . 
4 

The  last  two  cases  combined, 


Sometimes  the  resistance  of  the  web  's  omitted  from 
consideration.     In  such  a  case  the  intensity  of  stress  in 


664  ROLLED  AND   CAST  FLANGED  BEAMS.  [Ch.  XIV. 

each  flange  is  assumed  to  be  uniform  and  equal  to  either 
T  or  C.  At  the  same  time  the  lever-arms  of  the  different 
fibres  are  taken  to  be  uniform,  and  equal  to  h  for  one  flange 
and  h^  for  the  other,  h  and  h^  now  representing  the  vertical 
distances  from  the  neutral  axis  to  the  centres  of  gravity  of 
the  flanges,  while  d=h  +  hr 

On  these  assumptions,  if  /  is  the  area  of  the  upper  flange 
and  f  that  of  the  lower,  there  will  result 

M^fC.h  +  fT.h,.    .    ^.,>n,>     .     (5) 
But  since  the  case  is  one  of  pure  flexure, 

fC=f'T.      .  •:.,:  .     .    •;,.-  .     (6) 
...  M=fC(h  +  k1)=fCd=f'Td.  ><-V:'<,     (7) 

Also,  from  eq.  (6), 

/      T 


Or,  the  areas  of  the  flanges  are  inversely  as  the  ultimate 
resistances. 

Frequently  there  is  no  compression  flange,  the  section 
being  like  that  shown  in  Fig.  2.  In  such 
case  b  is  equal  to  bf,  or  tr  is  equal  to  zero; 
hence  b  =bf  in  eq.  (40),  but  no  other  change 
-  —  |  is  to  be  made  in  the  second  member  of  that 
-pIG  2  equation.  Eq.  (46)  may  then  be  used  pre- 

cisely as  it  stands  for  the  internal  resisting 
moment  of  a  beam  with  the  section  shown  in  Fig.  2. 

Prob.  i.  It  is  required  to  design  a  cast-iron  flanged 
beam  of  5  feet  effective  span  to  carry  a  load  of  1800  pounds 
applied  at  the  centre  of  span,  the  section  of  the  beam  to 
be  like  that  shown  in  Fig.  2,  i.e.,  without  upper  flange. 
The  greatest  permitted  working  stress  in  compression  will 


Art.  106.]          FLANGED  BEAMS   WITH  EQUAL   FLANGES.  66$ 

be  8000  pounds  per  square  inch,  and  the  total  depth  of  the 
beam  is  to  be  taken  at  9  inches. 

Referring  to  eqs.  (40),  (46),  and  Fig.  i  for  the  notation, 
the  given  data  and  the  dimensions  to  be  assumed  for  trial 
will  be  as  follows:  d  =  g  inches;  b=b'=l  inch;  bi=S 
inches;  ti  =  i  inch;  /  =  5  feet;  and  C  =  8ooo.  The  intro- 
duction of  these  values  into  eq.  (3)  will  give  for  the  distance 
of  the  centre  of  gravity  above  the  bottom  surface  of  the 
beam 

hi  =  2.6  inches     and     h  =d  —  hi  =  6.4  inches. 

The  preceding  trial  dimensions  will  make  the  beam 
weigh  about  50  pounds  per  lineal  foot.  If  all  the  preced- 
ing values  are  substituted  in  eqs.  (40)  and  (46),  remembering 

Wl 
that  M  =  — ,  there  will  be  found 

4 

W  •=  1994  —  125=  1869  pounds. 

The  trial  dimensions,  therefore,  give  the  centre-load 
capacity  of  the  beam  69  pounds  greater  than  required, 
which  may  be  considered  sufficiently  near  to  show  that  the 
assumed  dimensions  are  satisfactory. 

Art.  1 06. — Flanged  Beams  with  Equal  Flanges. 

Nearly  all  the  flanged  beams  used  in  engineering  prac- 
tice are  composed  of  a  web  and  two  equal  flanges.  It  has 
already  been  seen  that  the  ultimate  resistances,  T  and  C, 
of  structural  steel  and  wrought  iron  to  tension  and  com- 
pression are  essentially  equal  to  each  other ;  the  same  may 
be  said  a1  so  of  their  coefficients  of  elasticity  for  tension 
and  compression.  These  conditions  require  equal  flanges 
for  both  steel  and  wrought-iron  rolled  beams. 


666 


ROLLED  AND  CAST  FLANGED  BEAMS.          [Ch.  XIV. 


H— +— 
I 
I 


—B- 


In  Fig.  i  is  represented  the  normal  cross-section  of  an 
equal-flanged  beam.  It  also  approximately  represents 
what  may  be  taken  as  the  section  of  (  c 

any  wrought-iron   or   steel    I  beam,  the  * ftj * 

exact  forms  with  the  corresponding 
moments  of  inertia  being  given  in  hand- 
books. Although  the  thickness  t'  of  the 
flanges  of  such  beams  is  not  uniform, 
such  a  mean  value  may  be  taken  as 
will  cause  the  transformed  section  of 
Fig.  i  to  be  of  the  same  area  as  the 
original  section. 

Unless  in  exceptional  cases  where 
local  circumstances  compel  otherwise, 
the  beam  is  always  placed  with  the  web  vertical,  since  the 
resistance  to  bending  is  much  greater  in  that  position. 
The  neutral  axis  HB  will  then  be  at  half  the  depth  of  the 
beam.  Taking  the  dimensions  as  shown  in  Fig.  i ,  the  mo- 
ment of  inertia  of  the  cross-section  about  the  axis  HB  is 


FIG.  i. 


7  _ 


12 


while  the  moment  of  inertia  about  CD  has  the  value 


12 


(i) 


(2) 


With  these  values  of  the  moment  of  inertia,  the  genera] 

formula,  M=-r,  becomes  (remembering  that  d\=-    or    - 
di  \  22 


(3) 


or 


6b 


(4) 


Art.  106.]          FLANGED  BEAMS  WITH  EQUAL  FLANGES.  667 

k  is  written  for  all  extreme  fibre  stress. 

Eq.  (3)  is  the  only  formula  of  much  real  value.  It  will 
be  found  useful  in  making  comparisons  with  the  results 
of  a  simpler  formula  to  be  immediately  developed. 

Let    di  =  %(d+h).     Since    t'    is    small    compared    with 

-,  the  intensity  of  stress  may  'be  considered  constant  in 

2 

each  flange  without  much  error.  In  such  a  case  the  total 
stress  in  each  flange  will  be  kbt'  =  Tbt',  and  each  of  those 
forces  will  act  with  the  lever-arm  %di.  Hence  the  moment 
of  resistance  of  both  flanges  will  be 

kbt'-di. 

th* 
The  moment  of  inertia  of  the  web  will  be  —  .     Conse- 

quently its  moment  of  resistance  will  have  very  nearly  the 
value 

kth2 
6  ' 

The  resisting  moment  of  the  whole  beam  will  then  be 

(5) 


A  further  approximation  is  frequently  made  by  writing 
dji  for  H2\  then  if  each  flange  area  bt'  =/,  eq.  (5)  takes  the 
form 


il*Uh(f*f) (6) 

Eq.  (6)  shows  that  the  resistance  of  the  web  is  equivalent 
to  that  of  one  sixth  the  same  amount  concentrated  in  each 
flange. 


668  ROLLED  AND   CAST  FLANGED  BEAMS.          [Ch.  XIV 

If  the  web  is  very  thin,  so  that  its  resistance  may  be 
neglected, 

M  =  kfd1=kbt'di,     ......     (7) 

or 


Cases  in  which  these  formulae  are  admissible  will  be 
.given  hereafter.  It  virtually  involves  the  assumption  that 
the  web  is  used  wholly  in  resisting  the  shear,  while  the 
flanges  resist  the  whole  bending  and  nothing  else.  In 
other  words,  the  web  is  assumed  to  take  the  place  of  the 
neutral  surface  in  the  solid  beam,  while  the  direct  resistance 
to  tension  and  compression  of  the  longitudinal  fibres  of  the 
latter  is  entirely  supplied  by  the  flanges. 

Again  recapitulating  the  greatest  moments  in  the  more 
commonly  occurring  cases: 

Cantilever  uniformly  loaded, 

Wl    pP 
M  =  —  =  —  .......       (9) 

2         2 

Cantilever  loaded  at  the  end, 

M  =  Wl  .........     (10) 

Beam  supported  at  each  end  and  uniformly  loaded, 

Wl    pi2 

M=^--w  .....  11  (II> 

Beam  supported  at  each  end  and  loaded  at  centre, 

Wl 
M  =—  .  .     .     ......     (12) 

Beam  supported  at  each  end  and  loaded  both  uniformly 
and  at  centre, 


Art.  107.]  ROLLED  STEEL  FLANGED  BEAMS.  669 


In  all  cases  W  is  the  total  load  or  single  load,  while  p,  as 
usual,  is  the  intensity  of  uniform  load,  and  /  the  length  of  the 
beam. 

Art.  107.  —  Rolled  Steel  Flanged  Beams. 

The  resisting  moments  of  all  rolled  steel  beams  sub- 
jected to  bending  are  computed  by  the  exact  formula 


k  being  the  greatest  intensity  of  stress  (i.e.,  in  the  extreme 
fibres)  at  the  distance  dl  from  the  neutral  axis  about  which 
the  moment  of  inertia  /  is  taken.  In  all  ordinary  cases 
the  webs  of  beams  are  vertical  so  that  the  axis  for  /  is 
horizontal;  but  it.  sometimes  is  necessary  to  use  the  mo- 
ment of  inertia  /  computed  about  the  axis  passing  through 
the  centre  of  gravity  of  section  and  parallel  to  the  web. 
The  latter  is  frequently  employed  in  considering  the  lateral 
bending  effect  of  the  compression  in  the  upper  flange. 

The  upper  or  compression  flange  of  a  rolled  beam 
under  transverse  load,  unless  it  is  laterally  supported,  is 
somewhat  in  the  condition  of  a  long  column  and,  hence, 
tends  to  bend  or  deflect  in  a  lateral  direction.  This  ten- 
dency depends  to  some  extent  on  the  ratio  of  the  length  of 
flange  (/)  to  the  radius  of  gyration  (r)  of  the  section  about 
the  axis  parallel  to  the  web,  as  will  be  shown  in  detail  in 
a  later  article.  It  will  be  found  there  that  the  ultimate 
compression  flange  stress  decreases  as  the  ratio  l  +  r  in- 
creases. Hence  in  Table  I  there  will  be  found  values  of 
l  +  r  for  the  different  beams  tested.  • 


670  ROLLED  AND  CAST  FLANGED  BEAMS.  [Ch.  XIV. 

The  results  of  tests  given  in  Table  I  were  found  by 
Mr.  James  Christie,  Supt.  of  the  Pencoyd  Iron  Co.,  and 
they  are  taken  from  a  paper  by  him  in  the  "  Trans.  Am. 
Soc.  C.  E."  for  1884.  All  beams,  both  I  and  bulb,  were 
loaded  at  the  centre  of  span.  Hence  the  moment  of  the 
centre  load,  W,  and  the  uniform  weight  of  the  beam  itself, 
pi,  will  be,  as  shown  in  eq.  (13)  of  Art.  106, 


'— (2) 

4  \          '2  /      d\ 
Hence 


The  known  data  of  each  test  will  give  all  the  quanti- 
ties in  the  second  member  of  eq.  (3).  The  two  columns 
of  elastic  and  ultimate  values  of  k  in  the  table  were  com- 
puted by  eq.  (3).  The  positions  of  the  bulb  beams  (i.e., 
the  bulb  either  up  or  down)  in  the  tests  are  shown  by  the 
skeleton  sections  in  the  second  column. 

The  coefficients  of  elasticity  E  were  computed  from 
the  data  of  the  tests  taken  below  the  elastic  limit  by  the 
aid  of  eq.  (21),  Art.  28: 


(4) 


W  being  the  centre  load  and  pi  the  weight  of  the  beam, 
the  length  of  span  /  being  given  in  inches. 

All  beams  were  rolled  at  the  Pencoyd  Iron  Works. 
The  "mild  steel"  contained  from  o.n  to  0.15  per  cent,  of 
carbon,  and  the  "high  steel"  about  0.36  per  cent,  of  carbon. 
These  steels  are  the  same  as  those  referred  to  in  Art.  60. 

No.  14  is  the  only  test  of  a  "high"  steel  beam;   all  the 


Art.   107.] 


ROLLED  STEEL   FLANGED  BEAMS. 


67I 


remaining  tests  being  with  mild-steel  shapes.  Tests  3  to 
9  inclusive  were  of  deck  or  bulb  beams,  as  the  skeleton 
sections  show. 

Beams  3  and  4  were  rolled  from  the  same  ingot,  as  were 
also  6  and  7,  as  were  also  10,  12,  and  13,  and  as  were  also 
1 6,  17,  1 8,  and  19.  All  beams  were  unsupported  laterally 
in  either  flange.  The  moments  of  inertia  were  computed 
from  the  actual  beam  sections.  The  length  of  span  is 
represented  by  /,  while  r  is  the  radius  of  gyration  of  each 
beam  section  about  an  axis  through  its  centre  of  gravity 
and  parallel  to  its  web.  The  values  of  r  were  as  follows: 

5inch  I  .  . .  .r=o.54inch.  3  inchl.  . .  .r=o.59  inch. 


"  .  . .  .^=0.63 
"....r=o.7i 
" r=o.83 


8 
10 

12 


1 r=o. 


=0-95    ' 
=  1.01     " 


TABLE  I. 
TRANSVERSE  TESTS  OF  STEEL  BEAMS. 


No. 

Kind  of 
Beam. 

Span 
in  Ins. 

/  ^ 
r 

Moment 
of 
Inertia. 

Final 
Centre 
Load  in 
Pounds. 

k  in  Pounds  per 
Square  Inch  at 

Coefficient  of 
Elasticity  E, 
in  Pounds  per 
Square  Inch. 

Elastic. 

Ultimate 

i 

Mild    3"! 

59 

100 

2.76 

5,5oo 

41,100 

45,200 

30,890,000 

2 

3"  " 

39 

66 

2.76 

8,300 

40,800 

45,ioo 

25,01  1  ,000 

3 

"      s"! 

108 

200 

12 

8,800 

50,000 

55,ooo 

27,718,000 

4 

"        5"  i 

108 

2OO 

12 

8,400 

46,900 

52,500 

25,489,000 

5 

6    I 

96 

152 

22 

14,860 

51,200 

54,300 

23,692,000 

6 

7     " 

69 

97 

37.6 

34,000 

47,100 

59,300 

18,765,000 

7 

7     X 

69 

97 

37-6 

34,000 

47,100 

59,300 

23,040,000 

8 

9    I 

240 

290 

84.8 

14,500 

46,000 

51,300 

29,923,000 

9 

9    I 

240 

290 

82.9 

13-500 

39,800 

48,800 

30,209,000 

10 

81 

240 

273 

70.2 

13,000 

37,600 

44,400 

28,889,000 

ii 

8    " 

240 

273 

70.3 

12,930 

37,500 

44,100 

29,055,000 

12 

8    ' 

144 

164 

70.2 

19,480 

32,800 

39,900 

31,313,000 

13 

8    ' 

96 

109 

70.  2 

31,300 

40,300 

42,800 

23,689,000 

14 

Hgh    3    ' 

39 

2-74 

11,500 

54,3oo 



27,515,000 

IS 

M  Id  10    ' 

156 

164 

150.5 

22,500 

35,ooo 



28,414,000 

16 

10       ' 

1  68 

177 

150.5 

21,000 

35,200 

• 

27,182,000 

17 

'        IO      ' 

180 

189 

150.5 

19,500 

35,ooo 

. 

29,160,000 

18 

'        10      ' 

192 

202 

I50.S 

l8,000 

34,400 

29,727,000 

19 

'        12* 

240 

238 

264.7 

24,500 

33,400 



30,749,000 

20 

'        12* 

240 

238 

267.6 

24,200 

32,500 



29,568,000 

21 

'        12* 

228 

226 

273-8 

22.OOO 

27,500 



29,164,000 

22 

'         12       ' 

216 

214 

263.7 

29,000 

35,600 

30,219,000 

23 

'        12       ' 

204 

202 

256.7 

27  ,OOO 

32,100 



30,030,000 

24 

'        12       ' 

192 

190 

257-8 

34,ooo 

38,000 



29,709,000 

25 

'        12       ' 

190 

262.6 

34,000 

37,300 



28,234,000 

26 

'        12       ' 

1  80 

178 

262.4 

36,700 

37,7oo 



27,717,000 

27 

'        12       ' 

1  68 

166 

264.0 

38,000 

36,300 

28,784,000 

.28 

12* 

156 

154 

261  .  7 

43,000 

38,400 



27,818,000 

672  ROLLED  AND    CAST  FLANGED   BEAMS.         [Ch.  XIV. 

The  values  of  k  both  for  the  elastic  limit  and  the  ulti- 
mate are  erratic,  and  the  range  of  results  in  the  table  is  not 
sufficient  to  establish  any  law,  but  on  the  whole  the  small 
ratios  l-s-r  accompany  the  larger  values  of  k.  The  bulb 
or  deck  beams  also  appear  to  give  larger  values  of  k  than 
the  I  beams. 

The  results  of  these  tests  indicate  that  the  greatest 
working  intensities  of  stress  in  the  flanges  of  rolled  steel 
beams  may  be  taken  from  12,000  to  16,000  pounds  per 
square  inch  if  the  length  of  unsupported  compression 
flange  does  not  exceed  i5or  to  20or. 

In  the  work  of  design,  the  quantity  1-^-d^  used  in  eq.  (2), 
called  the  "section  modulus,"  is  much  employed,  and  it 
can  be  taken  directly  from  the  Cambria  Steel  Company's 
tables  at  the  end  of  the  book,  as  can  the  moment  of  inertia  /. 
Eq.  (2)  shows  that 

---  (s) 

d,~k  ' 

Hence  the  moment  of  the  loading  in  inch-pounds  di- 
vided by  the  allowed  greatest  flange  stress  in  pounds  per 
square  inch  must  be  equal  or  approximately  equal  to  the 
section  modulus  of  the  required  beam. 

There  may  be  found  in  the  Proceedings  of  the  American 
Society  for  Testing  Materials,  1909,  the  results  of  tests 
of  rolled  I  beams  and  girders  produced  by  the  Bethlehem 
Steel  Company  and  of  standard  rolled  I  beams  by  Profes- 
sor Edgar  Marburg.  Also  Professor  H.  F.  Moore  gives 
results  of  his  testing  of  steel  I  beams  of  the  regular  or 
standard  pattern  in  Bulletin  No.  68  of  the  University  of 
Illinois.  Professor  Marburg's  main  purpose  appears  to 
have  been  to  make  comparative  tests  of  the  ordinary 
I  beam  and  of  the  wide-flange  Bethlehem  shapes,  while 
the  principal  object  of  Professor  Moore  was  to  investigate 


Art.   107.] 


ROLLED  STEEL  FLANGED  BEAMS. 


673 


the  influence  of  lateral  deflection  on  the  capacity  of  the 
compressive  flange  without  lateral  support.  Table  II 
gives  the  results  of  these  tests,  each  of  professor  Marburg's 
results  except  one  being  an  average  of  three. 

TABLE  II. 
TESTS  OF  ROLLED  STEEL  BEAMS. 


Type. 

Span,  I 
Ft. 

I 
r' 

Extreme  Fibre  Stress 
k,  Lbs.  per  Sq.in. 

Modulus  of 
Elasticity. 

Size. 

Elas. 
Limit. 

Ultimate. 

Beth.  I.  . 

15 
15 
15 
15 
15 
15 
20 
20 
20 
20 
20 

125 
I67 

75 

125 
167 

75 
129 
176 
90 
in 
84 

31,700 
'  2O,4OO 
26,700 
21,800 
20,600 
22,500 
20,900 
19.500 
15400 
13,000 
1  1,  8OO 

46,100 
42,200 
53,900 
37,900 
34,700 
4I,IOO 
34,600 
33,000 
34,300 
32,300 
31,000 

26,900,000 
26,200,000 
26,900,000 
26,400,000 
26,900,000 
27,2OO,OOO 
26,400,000 
25,800,000 
25,600,000 
29,400,000 
24,800,000 

15'      38  Ib. 
15'      42 
15'      73  ' 
15'      38  ' 
15'      42 
15'      73 
24'      72 
24'      80 
24'    120 

30'      120 

30'  175 

Std.  I  

Girder 

Beth  I. 

Std.  I  
Girder  
Beth.  I  
Std.  I  
Girder  
Beth.  I*.  .  .. 
Girder 

*  One  beam  only. 

Prof.  Moore's  fifteen  tests  were  with  8-inch,  i8-pound  and  one  25-pound  I  beams, 
the  spans  being  5,  7.5,  7.92,  10,  15,  15,7  and  20  feet.  The  ratio  l-s-r'  varied  from  71  to 
286.  The  ultimate  fibre  stress  k  was  Max.  36,600;  Mean  32,300;  Min.  28,100.  The 
Modulus  E  was,  Max.  32,300,000;  Mean  28,400,000;  Min.  25,100,000.  The  Max.  E  is 
to  be  regarded  with  doubt. 

As  is  the  case  with  all  tests  of  full-size  rolled  beams, 
the  results  are  seen  to  vary  quite  widely.  This  is  largely  due 
to  the  fact  that  such  full-size  members  are  seldom  true 
in  all  their  parts,  i.e.,  the  web  may  be  a  little  twisted  on  the 
cooling  bed  and  the  flange  will  perhaps  never  be  perfectly 
plane,  consequently  the  applied  load  in  the  testing  machine 
will  not  be  received  with  presupposed  exactness.  Again, 
the  work  of  the  rolls  and  the  effects  of  cooling  will  not  be 
uniform.  At  any  rate  the  most  scrupulous  care  in  testing 
will  not  prevent  many  erratic  results,  apparently  unac- 
countable. 

In  order  to  show  these  results  graphically  they  have 


674 


ROLLED  AND   CAST  FLANGED   BEAMS.        [Ch.  XIV. 


been  plotted  on  Plate  I  and  the  explanatory  matter  on 
the   Plate  will  make  clear  the  results  belonging  to  each 

investigator.     The  horizontal  ordinate  is  the  ratio  — ,    r' 

being  the  radius  of  gyration  of  the  normal  section  of 
the  column  about  a  vertical  axis  parallel  to  the  web  and 
passing  through  its  centre.  The  vertical  ordinate  is  the 
intensity  k  of  the  extreme  fibre  stress  produced  by  the 
ultimate  load  on  the  beam  as  shown  in  Table  I. 
The  equation 

£=39,000-44—, 
represents  the  broken  line  drawn  on  Plate  I.     It  is  a  tenta- 


» 

PI 

itel 

^o-ooc 

•1 

• 

. 

0 

* 

40-000 

•1 

• 

X 

: 

"•" 

0 

-5" 

7 

~x 

30000 

•1 

•1 

• 
0 

•»  —  - 

T 

—  --X 

—  ^. 

T 

•y 

T 

20  000 

10000 

r—'LaJ^- 

X    Standard   I 
—    Moore 
•    Christie 
fc=39000  -<*•£, 

80 


120 


140 


180 


200 


220 


tive  expression,  as  there  are  not  sufficient  tests  with  the 
requisite  variation  of  —  to  justify  more  than  a  trial  value  of  k. 


Art.  107.]  ROLLED  STEEL  FLANGED  BEAMS.  675 

Professor  Marburg  made  no  effort  to  give  lateral  sup- 
port to  his  beams  under  test,  nor  did  he  endeavor  to  give 
the  compressive  flange  lateral  freedom,  as  did  Professor 
Moore  for  a  part  of  his  tests.  As,  however,  the  results 
appear  to  be  about  the  same,  whether  the  compressive  flange 
has  complete  lateral  freedom  or  not,  under  ordinary  cir- 
cumstances of  testing,  no  distinction  is  made  on  this  account 
between  the  various  plot  tings  on  Plate  I.  The  extremely 
high  values  on  that  Plate  belong  to  the  first  nine  tests 
by  Mr.  Christie,  as  given  in  Table  I.  They  are  abnormally 
high  and  whether  such  results  are  characteristic  of  bulb 
sections  or  due  to  some  other  reason  is  not  clear. 

Prob.  i.  It  is  required  to  design  a  rolled  steel  beam 
for  an  effective  span  of  20  ft.  to  carry  a  uniform  load  of 
725  Ibs.  per  linear  foot  in  addition  to  the  weight  of  the  beam 
itself,  the  circumstances  being  such  that  it  is  not  advis- 
able to  use  a  greater  total  depth  of  beam  than  12  ins. 
The  greatest  permitted  extreme  fibre  stress  k  will  be 
taken  at  12,000  Ibs.  per  sq.  in.  It  will  be  assumed  for 
trial  purposes  that  the  beam  itself  will  weigh  35  Ibs.  per 
linear  foot,  so  that  the  total  uniform  load  will  be.  760  Ibs. 
per  linear  foot.  The  centre  moment  in  inch-pounds  will, 
therefore,  be 

760X20X20X12 
M=—         —5—        — =456,000  in. -Ibs. 

o 

By  eq.  (5)  the  section  modulus  will  be  456,000-^12,000 
=  38.  By  referring  to  the  tables  in  almost  any  steel  com- 
pany's handbook  it  will  be  found  that  this  section  modulus 
belongs  to  a  1 2-inch,  35-pound  steel  rolled  beam,  and 
that  beam  fulfills  the  requirements  of  the  problem. 

Prob.  2.  It  is  required  to  design  a  rolled-steel  beam 
for  a  3  2 -ft.  effective  span  to  carry  a  load  of  1280  pounds  per 
linear  foot  in  addition  to  the  weight  of  the  beam,  and  a 


676  ROLLED  AND  CAST  FLANGED  BEAMS.          [Ch.  XIV. 

concentrated  load  of  1  1  ,000  pounds  at  a  point  1  1  feet  distant 
from  one  end  of  the  span.  The  greatest  permitted  work- 
ing stress  in  the  extreme  fibres  of  the  beam  is  16,000  Ibs. 
per  sq.  in. 

It  will  be  assumed  for  trial  purposes  that  a  24-in.  beam 
weighing  95  Ibs.  per  linear  foot  will  be  required  so  that 
the  total  uniform  load  per  linear  foot  will  be  1375  pounds. 
It  will  then  be  necessary  to  ascertain  at  what  point  in  the 
span  the  maximum  bending  moment  occurs,  i.e.,  at  what 
point  the  transverse  shear  is  equal  to  zero.  Let  a  be  the 
distance  of  the  concentrated  weight  from  the  nearest  end 
of  the  span,  i.e.,  a  =  n  ft.  Then  1et  P  be  the  single  weight, 
p  the  total  uniform  load  per  linear  foot,  and  /  the  length 
of  span.  The  following  equation  representing  the  condi- 
tion that  the  transverse  shear  must  be  equal  to  zero  may 
be  written 

pi  Pa 

--px  +  -r=o. 

I     Pa 
Hence  *  =  J+^T 

• 

In  the  above  equation  x  is  obviously  the  distance  from 
that  end  of  the  span  farthest  from  P  to  the  section  of 
greatest  bending  moment.  Substituting  the  above  numeri- 
cal values  in  the  equation  for  x,  there  will  result 

#  =  16  +  2.75  =  I&'7$  ft. 

Since  32  —  18.75=13.25  the  following  will  be  the  value 
of  the  greatest  bending  moment  in  inch-pounds  : 


1375X18.  75  11,000X1 

^-       -Xi3.25  +  --  -  -  -Xi.i2 

2,900,363  inch-pounds. 


1  \ 

-Xi8.75J 


Art.  108.]  DEFLECTION  OF  ROLLED  STEEL[  BEAMS.  677 

The  section  modulus  of  the  beam  required  is  by  eq.  (5) 
2,900,363-^16,000  =  181.  The  section  modulus  of  a  24. -in. 
steel  beam  weighing  85  Ibs.  per  linear  foot  is  180.7,  as  will 
be  found  by  referring  to  the  tables  at  the  end  of  the  book. 
Hence  that  beam  will  be  assumed  for  the  correct  solution 
of  the  problem.  The  fact  that  the  beam  weighs  10  Ibs. 
per  linear  foot  less  than  the  assumed  weight  has  too  small 
an  effect  upon  the  greatest  bending  moment  to  call  for 
any  revision. 

Prob.  3.  A  steel  tee  beam  of  8  ft.  span  is  to  be  used  as 
a  purlin  to  carry  a  uniform  load  of  125  Ibs.  per  linear  foot 
with  the  web  of  the  tee  in  a  vertical  position.  The  greatest 
permitted  intensity  of  stress  in  the  extreme  fibre  of  the 
tee  is  14,000  Ibs.  per  sq.  in.  It  is  required  to  find  the 
dimensions  of  the  tee.  By  referring  to  eq.  (5)  the  section 
modulus  will  be  written 

1000X96  . 

S  =  s--  -  =  .  86  in. 

8X  14,000 

By  referring  again  to  the  steel  handbook  tables  it 
will  be  found  that  a  3X3X1^  in.  steel  tee  weighing  6.6 
Ibs.  per  lin.  ft.  has  just  the  section  modulus  required. 
That  tee  therefore  fulfils  the  requirements  of  the  problem. 

Prob.  4.  It  is  required  to  support  a  single  weight  of 
12,000  Ibs.  at  the  centre  of  a  span  of  13  ft.  6  ins.  on  two 
rolled  steel  channels  with  their  webs  in  a  vertical  position 
and  separated  back  to  back  by  a  distance  of  3  ins.,  the 
greatest  permitted  intensity  of  stress  in  the  extreme  fibre 
of  the  flanges  being  15,000  Ibs.  Find  the  size  of  channels 
required. 

Art.  108.— The  Deflection  of  Rolled  Steel  Beams.      , 

The  deflections  of  rolled  steel  beams  may  readily  be 
computed  by  the  formulae  of  Art.  28.  The  general  pro- 


678  ROLLED  AND  CAST  FLANGED  BEAMS.  [Ch.  XIV. 

cedure  will  be  illustrated  by  using  the  equations  for  a  non- 
continuous  beam  simply  supported  at  each  end  and  loaded 
by  a  weight  at  the  centre  of  span,  or  uniformly,  or  in  both 
ways  concurrently.  Eq.  (20)  will  give  the  deflection  at 
any  point  located  by  the  coordinate  x,  while  eq.  (21)  will 
give  the  centre  deflection  only.  The  tangent  of  the  in- 
clination of  the  neutral  surface  at  any  point  located  by  x 

will  be  given  by  the  value  of  -7-  found  in  eq.  (19). 

Prob.  i.  Let  the  centre  deflection  of  the  rolled-steel 
beam  of  Prob.  i  of  Art.  107  be  required.  Referring  to 
eq.  (21)  of  Art.  22, 

W  =  o  ;     /  =  20  feet  =  240  inches  ;     p  =  760  pounds  ; 
7  =  228.3;     and     E  may  be  taken  at  29,000,000. 

Hence  the  centre  deflection  is 

_  240  X  240  X  240  X  5  X  760  X  20  _  . 

W*  ~  "48X8X29,000,000X228.3"  : 

If  half  the  external  uniform  load  of  725  pounds  per 
linear  foot  had  been  concentrated  at  the  centre  of  span, 

'°  =  7250  pounds;     p  =  35 


and  /  =  20  ft.  =  240  ins.     Also     pi  =  700  pounds. 

Hence  the  centre  deflection  would  be 

240  X  240  X  240  X  (7250  +  437  -5) 
^  =         48X29,000,000X228.3  =  '  333  lnch' 

Prob.  2.  In  Prob.  2  of  Art.  107  place  the  n,ooo-pound 
weight  at  the  centre  of  span,  then  find  the  inclination  of 
the  neutral  surface  and  the  deflection  of  the  24-inch  85- 


Art.   109:]  DROUGHT-IRON  ROLLED   BEAMS.  .  679 

pound  steel  beam  at  the  centre  and  quarter  points  of  the 
32-foot  span,  taking  £  =  29,000,000  pounds. 

Art.  109. — Wrought-iron  Rolled  Beams.      . 

Although  wrought-iron  rolled  beams  are  not  now  manu- 
factured, being  completely  displaced  by  steel  beams,  yet 
many  are  still  in  use.  Hence  it  is  advisable  to  exhibit 
the  empirical  quantities  required  to  design  them  and  to 
determine  their  safe  carrying  capacities  as  welt  as  their 
deflections  under  loading. 

It  has  been  observed  in  Art.  107  that  the  upper  or  com- 
pression flange  of  a  loaded  flanged  beam  will  deflect  or 
tend  to  deflect  laterally  at  a  lower  intensity  of  compressive 
stress  as  the  unsupported  length  of  such  a  flange  is  in- 
creased. The  experimental  results  given  in  Table  I  ex- 
hibit the  values  of  the  intensity  of  stress  K  in  the  extreme 
fibres  of  the  beam  both  at  the  elastic  and  ultimate  limits, 
the  usual  formula  for  bending  resistance  being  used, 


In  the  autumn  of  1883  an  extensive  series  of  tests  of 
wrought-iron  rolled  beams,  subjected  to  bending  by  centre 
loads,  was  made  by  the  author,  assisted  by  G.  H.  Elmore, 
C.E.,  at  the  mechanical  laboratory  of  the  Rensselaer  Poly- 
technic Institute.  The  object  of  these  tests  was  to  dis- 
cover, if  possible,  the  law  connecting  the  value  of  K  for 
this  class  of  beams  with  the  length  of  span  when  the  beam 
is  entirely  without  lateral  support.  The  means  by  which  the 
latter  end  was  accomplished,  and  a  full  detailed  account 
of  the  tests  will  be  found  in  Vol.  I,  No.  i,  "  Selected  Papers 
of  the  Rensselaer  Society  of  Engineers."  The  main  results 
of  the  tests  are  given  in  Table  I.  All  the  tests  were  made 
on  6-inch  I  beams  with  the  same  area  of  normal  cross- 


68o 


ROLLED  AND    CAST   FLANGED  BEAMS. 

TABLE  I. 


[Ch.  XIV. 


Final 

fi 

Perm'nent 

Perm'nent 

E 

No 

Span, 
Feet. 

Centre 
Weight, 
Pounds. 

/ 
r 

Elastic 
Limit, 
Pounds. 

Ultimate, 
Pounds. 

Vertical 
Deflection, 
Inches. 

Lateral 
Deflection, 
Inches. 

Pounds  per 
Square  Inch. 

I 

/ 

4,060 

400 

27,726 

3  1,094 

0.14 



24,170,000 

2 

f2° 

4,2OO 

400 

29,623 

32,885 

0.30 



26,374,000 

3 

IrS 

4,390 

360 

28,264 

30,79! 

O.2 

0-5 

24,520,000 

4 

f  l8 

4,57° 

360 

28,264 

32,020 

o.  18 

0.4 

24,313,000 

5 

!  Tfi 

•  4,77° 

320 

26,564 

29,579 

0.28 

.OO 

25,771,000 

6 

r6 

5,270 

320 

29,596 

32,632 

0.48 

.25 

25,003,000 

7 

1™ 

6,130 

280 

3!»X9X 

33,049 

0.30 

.20 

26,082,000 

8 

(•14 

6,125 

280 

3i,l64 

33,023 

0.30 

.  10 

23,373,000 

9 

/    To 

7,161 

240 

30,221 

32,907 

0.35 

.08 

25,287,000 

10 

f12 

7,35° 

240 

3i,3i4 

33,8i7 

0.33 

.09 

24,O22,OOO 

ii 

9,255 

200 

33,o82 

35,358 

0.39 

.08 

25,115,000 

12 

10 

r° 

9,655 

2OO 

33,o82 

37,o64 

0.50 

•50 

24,218,000 

13 

I    8 

n,485 

1  60 

29,736 

35,oio 

0.30 

0.90 

2I,6lI,OOO 

14 

1 

11,980 

1  6O 

3i,936 

36,527 

0.29 

•05 

21,987,000 

15 

[    6 

18,300 

1  2O 

35,497 

4i,737 

0.605 

•  53 

23,040,000 

16 

[ 

18,145 

I2O 

36,617 

4i,396 

0.67 

.88 

20,935,000 

17 

f     , 

22,870 

IOO 

34,1  36 

43,434 

0.67 

•75 

22,023,000 

18 

f    5 

23,065 

IOO 

34,136 

43,8i3 

0.67 

•75 

25,272,000 

19 

1    A 

29,985 

80 

32,619 

45,532 

0.96 

1.70 

24,315,000 

20 

f    4 

28,585 

80 

32,619 

44,744 

0.60 

1.86 

21,275,000 

section  of  4.35  square  inches.  Actual  measurement  showed 
the  depth  d  of  the  beams  to  be  6.16  inches.  The  moment 
of  inertia  of  the  beam  section  about  a  line  through  its 
centre  and  normal  to  the  web  wras  7  =  24.336.  The  radius 
of  gyration  of  the  same  section  in  reference  to  a  line  through 
its  centre  and  parallel  to  the  web  was  r  =  o.6  inch.  /  was 
the  length  of  span  in  inches. 

If  M  is  the  bending  moment  in  inch-pounds,  W  the 
total  centre  load  (including  weight  of  beam),  and  K  the 
stress  per  square  inch  in  extreme  fibre,  the  following 
formulae  result: 

f    Wl 

.  ==  4  .      ... 

wld  ro 

8/ (3' 


Md        . 
—      and 


(2) 


Art.  109.] 


DROUGHT-IRON  ROLLED   BEAMS. 


681 


The  experimental  values  of  W,  I,  d,  and  /  inserted  in 
the  above  formula  give  the  values  of  k  shown  in  the  table. 
The  coefficient  of  elasticity,  E,  was  found  by  the  usual 
formula, 

77      Wl*  f  ^ 

E  =  4^j>    •••••••     (4) 

in  which  w  is  the  deflection  caused  by  W. 

The  full  line  is  the  graphical  representation  of  the  values 
of  k  given  in  Table  I.  Since  k  must  clearly  decrease  with 

Piaie  t. 


)000- 


the  length  of  span,  and  increase  with  the  radius  of  gyration 
of  the  section  about  an  axis  through  its  centre  and  parallel 
to  the  web  (the  latter,  of  course,  being  vertical),  k  has 
been  plotted  in  reference  to  l  +  r  as  shown.  No  simple 
formula  will  closely  represent  this  curve,  but  the  bioken 
line  covers  all  lengths  of  span  used  in  ordinary  engineering 
practice,  and  is  represented  by  the  formula 


/ 
51,000-75-. 


(5) 


For  railway  structures  the  greatest  allowable  stress 
per  square  inch  in  the  extreme  fibres  of  rolled  beams  may 
be  taken  at 

I 


=  10,000—  15—. 


(6) 


682  ROLLED  AND   CAST  FLANGED  BEAMS.  [Ch.  XIV, 

Values  of  k  taken  from  a  large  scale  plate,  like  Plate  I, 
are,  however,  far  preferable  to  those  given  by  any  formula. 

The  ultimate  values  of  k  given  in  Table  I  are  fairly 
representative  of  the  best  wrought-iron  I  beams.  The 
coefficients  of  elasticity  E  range  from  about  22,000,000  to 
about  25,000,000  pounds;  the  average  may  be  taken  about 
24,000,000  pounds. 

The  deflection  of  wrought-iron  beams  may  be  computed 
by  the  formula 

WP  . 


when  the  load  W  is  at  the  centre  of  the  beam.  In  the 
general  case  of  a  beam  carrying  the  centre  load  W  and 
the  uniform  oad  pi,  the  quantity  (W  +  §pl)  must  displace 
W  in  eq.  (7).  If  the  beam  carry  only  the  uniform  load  pi, 
W  in  eq.  (7)  must  be  displaced  by  \pl. 

If  it  is  desired  to  apply  the  law  expressed  in  eqs.  (5) 
and  (6)  to  mild-steel  beams,  the  second  members  of  those 
equations  may  be  multiplied  by  £  to  f  for  close  approxi- 
mations. 


CHAPTER  XV. 

PLATE  GIRDERS. 

Art.  no. — The  Design  of  a  Plate  Girder. 

A  PLATE  girder  is  a  flanged  girder  or  beam  built  usually 
of  plates  and  angles,  the  flanges  being  secured  to  the  web 
by  the  proper  number  of  rivets  suitably  distributed.  The 
flanges,  unlike  those  of  rolled  beams,  are  usually  of  vary- 
ing sectional  area,  although  occasionally  either  flange  may 
be  of  uniform  section  throughout  when  formed  ,  of  two 
angles,  or  two  angles  and  a  cover-plate.  Fig.  i  is  a  general 
view  of  a  plate  girder,  while  Figs.  2,  3,  4,  and  5  show 
some  of  the  general  features  of  design. 

The  total  length  of  a  plate  girder  is  materially  more 
than  the  length  of  clear  span  over  which  the  girder  is  de- 
signed to  carry  load.  Blocks  or  pedestals  of  masonry  or 
metal,  as  the  case  may  be,  support  the  ends  of  the  girders 
and  rest  on  the  masonry  or  other  supporting  masses  or 
members  carrying  the  girder  and  its  load.  The  distance 
between  the  centres  of  these 'blocks  or  pedestals  is  called 
the  effective  span  of  the  girder,  as  it  is  the  span  length 
which  must  be  used  in  computing  bending  moments, 
shears,  or  reactions.  Plate  girders  must  evidently  be: 
somewhat  longer  than  the  effective  span.  \In  the  Figs,  the 
relations  of  the  various  parts  at  the  end  of  the  plate  girder 
are  shown  in  detail.  The  girder  illustrated  in  Fig.  i  has 

683 


684  PLATE  GIRDERS.  [Ch.  XV. 

an  effective  span  of  68  ft.  with  the  centre  of  the  pedestal 
block  15  inches  from  the  face  of  the  masonry  abutment 
and  12  inches  from  the  extreme  end  of  the  girder.  The 
-  effective  depth  of  the  girder  is  '  the  vertical  distance  or 
depth  between  the  centres  of  gravity  of  the  two  flanges. 
When  the  girder  has  cover-plates  this  effective  depth  may 
be  greater  than  the  depth  of  web  plate  at  the  centre  of 
span  and  less  than  that  at  the  ends,  even  when  the  web 
plate  is  of  uniform  depth.  It  is  always  customary,  how- 
ever, to  take  the  effective  depth  of  a  plate  girder  with 
uniform  depth  of  web  as  constant.  Frequently  that  depth 
is  taken  equal  to  the  depth  of  the  web  plate;  or,  again,  it 
may  be  taken  equal  to  the  depth  between  the  centres  of 
gravity  of  the  flanges  at  mid-span  without  sensible  error. 
In  case  the  web  plate  is  not  of  uniform  depth  the  effective 
depth  might  still  be  taken  as  the  depth  of  web  plate  at 
the  various  sections  of  the  girder,  or  it  may  be  taken  as  the 
depth  between  centres  of  gravity  of  the  flanges  at  the  same 
sections. 

The  plate  girder  shown  in  Fig.  i  and  to  be  assumed  for 
the  purposes  of  design  is  of  the  deck  type  and  has  a  clear 
span  of  65  ft.  6  ins.,  an  effective  span  of  68  ft.,  and  a  length 
over  all  of  70  ft.  The  differences  between  the  effective 
span  and  the  clear  span  and  total  length  are  obviously 
dependent  upon  the  length  of  span.  For  short  spans 
those  differences  are  relatively  small,  and  relatively  large 
for  long  spans.  The  depth  of  web  plate  will  be  taken  as 
6  ft.  8  ins.,  and  it  will  be  found  later  that  at  and  in  the  vicin- 
ity of  the  centre  of  span  three  cover-plates  will  be  needed. 
The  girder  will  be  assumed  to  be  of  mild  structural  steel 
and  will  be  supposed  to  carry  a  single-track  railroad  mov- 
ing load  with  the  concentrations  and  spacings  shown  in 
Table  I,  Art.  21. 

The  dead  load  or  own  weight  of  the  girder  and  track 
will  depend  somewhat  upon  whether  the  girder  is  of  the 


Remainder  c 
as  Top  Flanj 
holes  to  be  f 


.notch  i  1 

to    i    for  flg.  L  Tight 

oiui 

c2" 

3 

V      IN" 

Eg 

eio^b.  tob./.8  | 

3! 

0 

£ 

"s 

it    e 

^f°                  'g 

—  G 

-9  

i 

i 

*o 

,      2  L'o'i  4*      ° 
Q   i  ?^"  x  7  10"  »    i 

-', 

Li 

2L85x  3>j  T  V* 

"S 

0 

2  Fills  3V  "  "V: 
<     «V««'^      J 

=• 

*s 

2JS  4^ 

a"  Fig.  5 

10  of  3=  26* 

l« 

%  3>4" 

3^" 

(To  face  page  63 S-) 


Art.   no.]  THE  DESIGN  OF  A  PLATE   GIRDER.  685 

through  or  deck  type.  The  only  difference  in  computa- 
tion arising  in  those  two  types  is  due  to  the  fact  that  if  the 
girders  are  of  the  deck  class  (i.e.,  carrying  the  moving  load 
directly  on  their  upper  flanges)  the  rivets  connecting  the 
upper  flanges  with  the  webs  must  be  assumed  to  carry  the 
wheel  concentrations  in  addition  to  their  other  duties,  as 
will  be  shown  in  the  following  computations.  The  total 
dead  load  or  own  weight  will  be  taken  as  1400  Ibs.  per  linear 
foot.  Inasmuch  as  there  are  two  girders,  each  will  carry 
one  half  of  the  moving  load  and  one  half  of  the  dead  load 
or  own  weight.  It  should  be  observed  that  the  effective 
length  of  span  being  68  ft.,  the  two  locomotives  at  the  head 
of  the  train  load  will  more  than  cover  the  span,  so  that  the 
uniform  train  load  will  not  appear  in  the  computations. 

The  design  of  this  plate  girder  will  be  made  in  accord- 
ance with  the  provisions  of  the  American  Railway  Engi- 
neering and  Maintenance  of  Way  Association  and  refer- 
ences will  be  made  to  those  provisions. 

Bending  Moments. 

The  first  computations  necessary  are  those  required 
to  determine  the  bending  moments,  and  from  them  the 
flange  stresses  at  different  points  of  the  span.  Those 
points  may  be  taken  at  5,  8,  or  10  ft.  apart  as  may  be 
desired  for  the  purpose  of  design;  the  closer  together  the 
sections  are  taken  the  greater  will  be  the  degree  of  accuracy 
attained.  In  the  present  instance  those  sections  will  be 
taken  5  feet  apart  up  to  25  ft.  from  the  end  of  the  span, 
but  the  next  or  final  section  will  be  at  the  centre  of  span. 
After  the  bending  moments  are  obtained,  the  flange 
stresses  at  once  result  by  dividing  the  former  by  the  effec- 
tive depth. 

Figs,   i  and   2  show  the  complete   single-track  railway 


686  PLATE   GIRDERS.  [Ch.  XV. 

deck-plate  girder  span  consisting  of  two  girders  with  the 
requisite  bracing  connections  between  them.  The  total 
dead  load  or  own  weight  is  a  uniform  load  and  consists  of : 

Lbs.  per  Lm.  Ft. 

Track  (ties,  rails,  etc.) 450 

Two  girders  and  bracing 1050 


Total  .....................  1500 

r^     £  '    J  ISOO 

Or  for  one  girder.  .  .  ............   ~  —  =    750 

2 

As  each  girder  will  carry  750  Ibs.  of  dead  load  per  linear 
foot,  and  as  the  effective  span  is  68  ft.,  the  expression  for 
the  dead-load  bending  moment  in  foot-pounds  at  any 
point  will  be  as  follows: 


(i) 


The  application  of  eq.  (i)  to  the  sections  of  the  girder 
5,  10,  15,  20,  25,  and  34  ft.  from  the  ends  will  give  the 
following  expressions  for  the  bending  moments  in  foot- 

pounds : 

D.  L.  Moment. 
x  ,      Ft.  Lbs. 

5  ...............................  118,120 

10  ..........  ......  .............  217,500 

15.  ......  ,..      ..  ...................  298,100 

20  .........................  ......  360,000 

25  ....................  ...........  403,100 

34  .............................  433,500 

The  moving-load  bending  moments  are  next  to  be  found 
by  using  the  concentrations  shown  in  Table  i,  Art.  21. 
For  this  purpose  the  criterion  for  the  maximum  bending 


Art.  I  io.]  THE  DESIGN  OF  A  PLATE   GIRDER.  687 

moment,  eq.  (5),  Art.  21,  must  be  applied  at  the  assumed 
sections  in  which  I'  (equal  to  oo  in  the  above  dead-load 
computations)  has  the  values  5,  io,  15,  20,  25,  and  34  ft. 
The  application  of  that  criterion  to  the  section  BO,  Fig.  i, 
5  ft.  from  the  end  of  the  span  shows  that  W2,  or  the  first 
driving  wheel,  must  rest  at  the  section  in  question  for  the 
maximum  bending  moment,  the  loads  Wi  to  Wi2  inclusive 
resting  on  the  span.  Wi  will  be  off  the  span.  By  the  aid 
of  Table  i,  Art.  21,  the  greatest  bending  moment  desired 
is: 

Ms  =-^-(9,030,000+2  X273,ooo)  =704,000  ft.-lbs. 
68 

Similarly  for  the  section  CN,  io  ft.  from  the  end  of  the 
span,  the  criterion  eq.  (5)  of  Art.  21  shows  that  W%  must 
be  placed  at  C  with  W\z  2  ft.  from  the  end  of  the  span 
and  Wi  off  the  span.  By  the  aid  of  Table  i  the  desired 
moment  takes  the  value: 

M 10  =77: (9, 030, 000  +  2  X273,ooo)  — 150,000=  1,260,000  ft.-lbs. 
68 

Concisely  stating  the  conditions  and  results  for  the 
remaining  sections  shown  on  Fig.  i :  For  DL,  1 5  feet 
from  end  of  span,  two  positions  of  moving  load,  T/F3  at  D 
ancb  Wi2  at  D  satisfy  the  criterion,  but  the  latter  with 
13  feet  of  uniform  train  load  on  the  span  gives  the  greatest 
moment.  Total  load  on  the  span  is 

(Wio+  .  .  .    +  1^18+3000X13) 
and  the  moment  is: 

j-/  T~2\ 

M}5  =7^(6,310,000  +  2 13,000 X 13  +3000  X— )  -345,000  = 

Oo  \  2    / 

1,715,000  ft.-lbs. 


688  PLATE   GIRDERS.  [Ch.  XV. 

For  EM,  20  feet  from  end  of  span,  place  Wi2  at  E\ 

M20  =^(6,3 10,000 +8  (2 13,000  H 3ooo\  \  _345>ooo== 

o8\  \  2        // 

•2,040,000  ft.-lbs. 

For  GH,  25  feet  from  end  of  span,  place  Wi2  at  G  and 
the  moment  is: 

M25  =^-(7,500,000  +  232, 500X3  +3000^-)  -755,000  = 

DO  \  2  / 

2,265,000  ft.-lbs. 

The  moment  at  the  centre  of  the  span  can  be  computed 
in  the  same  manner,  but  by  referring  to  Table  II  of  Art. 
21,  it  will  be  seen  to  be: 

M34  =  2,435,4oo  ft.-lbs. 

A  reference  to  the  American  Railway  Engineering 
and  Maintenance  of  Way  Association  specifications,  Art. 
9,  will  show  that  the  required  allowance  for  impact  is 
represented  by  the  factor  I,  in  which  L'  is  the  length  of 
load  on  the  span: 


1  = 


300 


I/+300* 


The  positions  of  loading  already  found  for  the  greatest 
moving  load  moments  give  the  lengths  L'  in  feet  in  the 
following  table: 


Pt. 

Ft. 

Loaded 
Length,  L'. 
Ft. 

Impact 
Factor  /. 

Moving  Load 
Moment,  Ft.-lbs.   « 

Impact  Moment 
—  ...    Ft.-lbs. 

5 

63 

.827 

7O4,OOO 

582,000 

10 

63 

.827 

1,260,000 

1,040,000 

15 

66 

.820 

1,715,000 

1,405,000 

20 

61 

•897 

2,040,000 

1,830,000 

25 

64 

•825 

2,265,000 

1,866,000 

34 

68 

.815 

2,435,000 

1,985,000 

Art.  no.]  THE  DESIGN   OF  A   PLATE  GIRDER.  689 

By  adding  the  'dead  load  or  own  weight  moments, 
already  computed,  to  the  moving  load  and  impact  moments 
in  the  preceding  table,  the  total  or  resultant  moments 
will  be: 

TABLE  I. 

-P,  Total  Moment 

"•  Ft.-lbs. 

5  .........................    1,404,000 

10  .........................  '  2,518,000 

15  .........................  3,418,000 

20  .........................  4,230,000 

25-  •'  .......................  4,534,000 

34  .........................  4,855,000 

Shears. 

Both  dead  and  moving  load  shears  must  be  computed. 
As  the  dead  load  or  own  weight  is  a  uniform  load  on  the 
girder,  the  shear  at  any  point  is  simply  the  load  between 
that  point  and  the  centre  of  span.  Hence  indicating  the 
transverse  shear  at  any  section  by  the  figure  showing  its 
distance  from  the  end  of  the  span,  there  will  result  the 
following  values,  50  being  the  end  shear  or  reaction: 

50=34X750  =  25,000  Ibs. 

.  • 

55  =  29X750  =  21,750 

510  =  24X750  =  18,000  " 

515  =  19X750  =  14,250  " 

520  =  14X750  =  10,500  " 

525=   9X750=   6,750  " 

=   0X750=          o  " 


The  moving  load  shears  will  also  be  needed.     Although 
there  is  no  systematic  criterion  for  such  shears  at  different 


690  PLATE  GIRDERS.  [Ch.  XV. 

points  in  a  span  traversed  by  a  train  of  concentrations, 
it  is  a  simple  matter  to  find  the  greatest  moving  load  shears 
at  the  sections  contemplated  by  inspection  and  trial.  The 
greatest  end  shear,  i.e.,  the  greatest  reaction,  has  been 
found  in  Art.  21  and  is  given  in  Table  II  of  that  Article: 

End  shear  for  68-ft.  span  =  161,700  Ibs. 
End  impact  shear  =131, 800    '  * 

The  impact  factors  for  the  shears  are  computed  by  the 
same  formula  already  used  for  impact  moments. 

For  a  shear  5  feet  from  end  of  span :  place  W%  at  the 
5 -foot  section,  then  the  greatest  shear  is 

c      9,030,000  +  2X273,000  1U 

55=-  -  =141,000  Ibs. 

Oo 

By  trying  other  positions  it  will  be  found  that  this 
gives  the  greatest  shear.  W\  is  not  on  the  girder  and  W\* 
is  2  feet  from  the  end  of  the  span. 

For  section  10  feet  from  end;  place  W\\  at  the  section. 
Hence 

6,310,000  +  213,000  Xi3  —  (3000  X— - 

5 10  = -150,000  = 

OO 

122,000  Ibs. 

For  section  15  feet  from  end:  place  W\\  at  the  section 
and  there  will  result 

82 
6, 3 10, ooo +  2 13, ooo  X 8 +3000  X  — 

5i5=-  — gg-  --150,000  = 

104,300  Ibs. 


Art.  no.] 


THE    DESIGN  OF  A  PLATE    GIRDER. 


691 


For  20-ft.  section:   place  W2  at  the  section  and  there 
will  result 


6,050,000  „ 

=  -  --  150,000=87,200  Ibs. 

Oo 


For  a  2  5  -ft.  section:   place  W%  at  the  section  and  the 
greatest  shear  will  be 

$,  240,  000+213.000  Xs 
525  =  —  —  -  1  50,000  =  7  1  ,  500  Ibs. 

Oo 


For  the  centre  of  span  :   place 
greatest  shear  will  be: 


at  that  point  and  the 


~        3,230,000  +  174,000X5  1U 

534=  -  -150,000=45,300  Ibs. 

08 


The  loaded  lengths  in  each  of  these  cases  to  be  used 
in  computing  the  impact  factors  are  in  the  order  of  the 
sections  beginning  with  that  at  5  feet  from  the  end,  63, 
66,  61,  56,  51,  and  42  feet,  the  latter  belonging  to  the 
centre  of  span.  The  following  tabular  statement  repre- 
sents the  elements  of  these  moving  load  shears  and  the 
impact  allowances: 


SHEARS  AND  IMPACT  ALLOWANCES 


Section. 

Loaded 

Length.     Ft. 

Impact 
Factor. 

Moving  Load 
Shear.     Lbs. 

Impact  Shear. 
Lbs. 

5 

63 

.827 

I4I,OOO 

116,500 

10 

66 

.820 

122,000 

100,000 

15 

61 

.831 

104,300 

86,600 

20 

56 

.824 

87,20O 

71,800 

25 

51 

•855 

71.500 

61,100 

34 

42 

•877 

45,300 

39,700 

692  PLATE  GIRDERS.  [Ch.  XV. 

Adding  together  the  dead  load,  moving  load  and  impact 
shears  as  now  determined,  the  following  will  be  the  resultant 
or  total  shears  at  sections  under  consideration: 

TABLE  II. 

RESULTANT  OR  TOTAL  SHEARS. 

Section.                                                                              lotal^hears. 
End 319,000 

5 279,300 

10 240,500 

15 205,100 

20 169,500 

25 139,400 

34 85,000 

The  preceding  results  or  computations  due  to  the  dead 
and  moving  loads  are  the  principal  data  required  in  the 
design  of  the  girder. 

Web  Plate. 

The  effective  depth  of  the  girder  will  tentatively  be 
taken  as  6  feet  8  inches  and  the  depth  from  the  back  of 
flange  angles  in  the  upper  flange  to  the  back  of  the  lower 
flange  angles  will  be  taken  as  6  feet  8J_  inches.  As  the 
depth  of  the  web  plate  must  be  taken  a  little  less  than  the 
depth  from  back  to  back  of  angles,  in  order  that  the  flange 
plates  may  not  touch  the  edges  of  the  web  plates  when 
the  different  parts  of  the  girder  are  assembled,  that  depth 
should  be  taken  as  6  feet  8  inches.  In  fact  the  effective 
depth  of  a  plate  girder  is  sometimes  prescribed  as  the  depth 
of  the  web  plate.  This  depth  of  web  plate  will  leave 
\  inch  clear  at  the  top  and  bottom  flanges,  which  is  sufficient 
to  insure  the  flange  plates  freedom  from  hitting  the  edges 
of  the  web. 

Art.  1 8  of  the  Specifications  allows  a  working  stress 
in  shear  of  10,000  pounds  per  square  inch  of  gross  cross- 


Art.  1 10.]  THE  DESIGN    OF  A  PLATE    GIRDER.  693 

section  of  the  web.  As  the  total  end  shear  has  been 
found  to  be  319,000  pounds,  the  gross  web  plate  section 
at  the  end  of  span  should  be  31.9  square  inches.  The 

minimum  thickness  must  then  be  ^^  =  .399  inch. 

80 

A  web  plate  80  X-^  inch  will  be  used,  giving  a  gross 

sectional  area  of  80  X. 43  7  5  =3  5  square  inches.  The  sur- 
plus area  is  small  and  it  is  judicious  design  to  have  it. 
This  web  plate  thickness  also  satisfies  Art.  29  of  the  Speci- 
fications which  prescribes  that  "  The  thickness  of  web 

plates  shall  not  be  less  than  — —  of  the  unsupported  dis- 

100 

tance  between  fiange  angles,"  as  6X6  inch  flange  angles 
will  be  used, 


Flanges. 

Art.  29  of  the  Specifications  provides  that  the  design 
of  the  flanges  may  be  based  either  on  the  moment  of  inertia 
of  the  net  section  of  the  girder  or  on  the  assumption  that 
the  flange  stress  is  of  constant  intensity  with  its  centre 
at  the  centre  of  gravity  of  the  flange  area,  the  latter 
including  one-eighth  of  the  gross  section  of  the  web,  the 
difference  between  one-sixth  and  one-eighth  -of  the  web 
section  being  supposed  to  cover  the  material  punched  out 
in  the  tension  side  of  the  web  plate.  The  latter  method 
will  be  employed. 

Art.  30  of  the  Specifications  provides  that  "  The  gross 
section  of  the  compression  flanges  of  plate  girders  shall 
not  be  less  than  the  gross  section  of  the  tension  flanges." 
It  will  be  best,  therefore,  to  design  the  tension  flange 
first. 

Using  the  total  or  resultant  bending  moment  at  the 


7 
694  PLATE  GIRDERS.  [Ch.  XV. 

centre  of  the  span,  the  trial  effective  depth  of  6  feet  8  inches 
will  give  the  centre  flange  stress  as  follows: 

4,855,000 

•        ,     -  =  728,000  Ibs. 
6.67 

The  specifications  permit  a  working  tensile  stress  in 
the  net  section  of  the  tension  flange  of  16,000  pounds  per 
square  inch.  Hence  the  required  net  tension  flange  area 
is 

728,000 


The  available  flange  section  due  to  one-eighth  the  gross 
sectional  area  of  the  web  is  ^  =4.375  square  inches.  The 

o 

amount  of  flange  area  to  be  supplied  by  the  flange  plates 
and  angles  is,  therefore, 

45.5-4.4=41.1  sq.ins. 

In  providing  41.1  square  inches  it  is  necessary  to  know 
what  rivet  holes  are  to  be  deducted  from  each  cover-plate 
and  each  flange  angle.  It  is  clear  that  two  rivet  holes 
only  need  be  deducted  from  each  cover-plate,  and  it  is  plain 
that  at  least  two  rivet  holes  must  be  deducted  from  each 
flange  angle  section.  In  designing  cover-plates  for  flanges 
it  must  be  remembered  that  no  such  plate  must  be  thicker 
than  the  one  under  it,  i.e.,  if  these  plates  are  not  of  the  same 
thickness,  the  thickest  one  must  lie  on  the  angles,  the 
remaining  thicknesses  to  decrease  or  be  the  same  in  passing 
outward  from  the  angles.  As  a  trial  section  let  the  follow- 
ing be  assumed: 


Art.  1 10.] 


THE  DESIGN   OF  A  PLATE  GIRDER 


695 


Angles  or  Cover-plates. 

Gross  Area. 
Sq.Ins. 

Less  Rivet  Holes. 
Sq.Ins. 

Net  Section. 
Sq.Ins. 

2  6"X6"X|"  
3  covers  14"  X  I".  ... 

* 

16.88 
31-5 

w        *       ^ 

4XlXf=3.0 

6XlX|=4-5 

13-88 
27.00 

48.38 

40.88 

As  40.88  square  inches  is  but  ij  per  cent,  less  than  the 
desired  area,  41.4  square  inches,  the  former  may  be  accepted 
subject  to  further  confirmation. 

If  the  centre  of  gravity  of  the  gross  section  of  the  tenta- 
tive flange  area  consisting  of  the  three  plates  and  two 
angles  indicated  above  be  determined,  it  will  be  found 
.  1 1  inch  above  the  back  of  the  angles.  This  will  make  the 
effective  depth 

6  ft.  8.5  ins.  +  .22  in.  =  6  ft.  8.72  ins. 

This  increase  in  effective  depth  will  correspondingly 
decrease  the  centre  flange  stress  so  as  to  make  the  total 
actual  net  area  of  45.3  square  inches  a  little  larger  than 
required.  Hence  the  trial  centre  tension  flange  area  as 
determined  above  will  be  accepted  as  the  actual  flange  area 
to  be  used,  i.e.,  three  i4Xf-inch  cover-plates  and  two 
angles  6X6X|  inch. 


Length  of  Cover-plates. 

In  the  next  Article  there  will  be  shown  two  methods 
of  determining  the  lengths  of  cover-plates  after  the 
sections  of  those  plates  have  been  found  for  the  greatest 
bending  moment,  usually  taken  as  at  the  centre  of  span. 
These  two  methods  are  simply  different  forms  of  expres- 
sion of  the  same  thing.  The  following  notation  will  be 
used: 


696  PLATE  GIRDERS  [Ch.  XV. 

/  =  length  of  span  in  feet ; 
Li  =  length  of  outside  cover-plate  in  feet; 
L2  =  length  of  second  cover-plate  in  feet; 

A  =  total  net  flange  area,  square  inches; 

ai  =net  area  of  outside  cover-plate,  square  inches; 

a2  =net  area  of  second  cover-plate,  square  inches; 

as  =net  area  of  third  cover-plate,  square  inches. 

It  has  already  been  seen  that  if  a  beam  simply  supported 
at  each  end  be  loaded  uniformly  throughout  the  span,  the 
bending  moment  at  any  point  will  be  represented  by  the 
vertical  ordinate  of  a  parabola  whose  vertex  is  over  the 
centre  of  span  while  the  end  of  each  branch  is  at  one  end 
of  the  span.  It  is  assumed  that  the  greatest  bending 
moments  in  the  plate  girder,  already  computed,  vary  by 
the  same  parabolic  law.  This  is  not  quite  true,  but  suf- 
ficiently near  for  ordinary  purposes. 

Then,  as  will  be  shown  in  the  next  Article, 


+a2.        T 

~    3 


# 


In  this  case  /  =68  feet  and  A  =45.3  square  inches. 
a.i  =  a2  =  03  =  9  sq.  ins. 

Making  these  numerical  substitutions,  there  will  result 
Li  =30.7  feet.;  Li  =42.9  feet;  L3  =  52.5  feet.  These  lengths 
are  clearly  the  minimum  permissible.  In  actual  construc- 
tion it  is  desirable  to  have  the  end  of  the  plate  from  i  to 
1.5  feet  further  from  the  centre,  making  the  total  length  of 
the  plate  2  to  2.5  feet  greater  than  the  length  computed 
above.  This  lengthening  of  the  cover-plate  is  essential 
in  order  that  the  cover-plate  metal  may  be  taking  stress 
at  the  point  where  the  plate  is  computed  to  begin.  Also 


Art.  no. 


THE  DESIGN    OF  A   PLATE  GIRDER. 


697 


as  will  be  seen  a  little  further  on,  the  pitch  of  rivets  in 
these  ends  of  the  cover-plates  is  made  less  than  in  the 
body  of  the  plate  for  greater  effectiveness  where  the  plate 
begins  to  take  its  stress.  The  lengths  of  cover-plates 
then,  beginning  with  the  shortest,  will  be  33.2,  45.4,  and 
55  feet. 

Another  method  of  procedure,  more  accurate  than  the 
preceding,  is  to  draw  a  moment  curve  on  the  effective 
span,  which  can  readily  be  done  by  laying  down  as  vertical 
ordinates  the  resultant  or  total  moments  as  given  in  Table  I. 
These  moment  ordinates  would  be  5  feet  apart  except 
at  the  centre  of  span.  The  lengths  of  cover-plates  must 
be  such  as  to  give  resisting  moments  of  the  flange  stresses 
at  least  equal  to  the  external  bending  moments  shown 
on  such  a  diagram.  The  moments  of  the  flange  stresses 
will  require  the  centres  of  gravity  of  parts  of  the  flange 
sections  to  be  computed  at  each  moment  point.  The 
following  tabulation  shows  the  elements  of  this  method 
of  procedure  for  the  centre  section  of  the  girder: 


Section. 

Sq.  Ins. 

Stress  per 
Sq.  In. 

Lever  Arm 
Ft. 

Moment. 
Ft.-lbs. 

One-eighth  web  plus  flange  angles  .  .  . 
First  cover-plate  

18-3 
Q 

l6,OOO 
l6,OOO 

6.41 

6    77 

1,875,000 

076  ooo 

Second  cover-plate. 

Q 

1  6  ooo 

6  81 

Top  cover-plate 

1  6  ooo 

6    Q 

yy4,uuu 

This  operation  must  be  repeated  at  each  moment  section 
of  the  girder,  but  the  numerical  work  need  not  be  repeated 
here,  being  precisely  like  that  for  the  centre  section. 

The  net  lengths  of  plates  found  by  this  method  are  32.8, 
42.9  and  53.8  feet,  a  substantial  agreement  with  the  lengths 
found  by  the  shorter  procedure. 

In  the  compression  flange  the  cover-plate  lying  on  the 
angles  should  run  the  entire  length  of  the  girder,  especially 


698  PLATE  GIRDERS.  [Ch.  XV. 

if  the  girder  be  of  the  deck  type,  i.e.,  with  ties  resting  upon 
the  upper  flange.  That  flange  being  under  compression, 
it  is  advisable  that  the  horizontal  legs  of  the  angles  be 
supported  throughout  their  entire  length  by  riveting 
them  to  a  cover-plate.  This  will  add  to  the  stiffness  and 
carrying  capacity  of  the  flange.  If  ties  rest  directly  upon 
the  upper  flange,  their  deflection  tends  to  bend  one  side 
of  it  out  of  its  horizontal  position,  but  this  tendency  will 
be  materially  lessened  by  the  added  stiffness  gained  in 
riveting  the  horizontal  angle  legs  of  the  flange  to  the  cover- 
plate. 

Although  this  process  of  design  has  been  used  in  con- 
nection with  the  tension  flange,  under  the  specifications 
the  compression  flange  is  to  be  made  like  the  tension  flange, 
i.e.,  a  duplicate  of  it. 

Pitch  of  Rivets  in  Flanges. 

Arts.  5  and  31  of  the  specifications  relate  to  the  rivets 
required  to  join  the  vertical  legs  of  the  flange  angles  to  the 
web  plate.  Art.  31  requires  that  "The  flanges  of  plate 
girders  shall  be  connected  to  the  web  with  a  sufficient 
number  of  rivets  to  transfer  the  total  shear  at  any  point 
in  a  distance  equal  to  the  effective  depth  of  the  girder  at 
that  point  combined  with  any  load  that  is  applied  directly 
on  the  flange.  The  wheel  loads  where  the  ties  rest  on  the 
flanges  shall  be  assumed  to  be  distributed  over  three  ties." 

The  chief  function  of  these  rivets  is  to  transfer  hori- 
zontal shear  from  the  web  plate  to  the  flanges,  as  it  is  in 
this  way  that  the  flanges  receive  their  stresses.  If  the 
rivets  take  the  direct  load  of  the  locomotive  driving  wheels, 
as  in  the  case  of  a -deck  girder  like  that  being  designed, 
they  must  resist  the  resultant  stress  due  to  both  vertical 
and  horizontal  loads. 


Art.  no.]  THE  DESIGN  OF  A   PLATE  GIRDER.  699 

Strictly  speaking  the  number  of  rivets  required  between 
two  moment  sections,  as  shown  in  Fig.  i,  should  be  just 
sufficient  to  give  the  increase  of  flange  stress  in  passing 
from  one  section  to  the  next  one  toward  the  centre  of  span. 
Art.  31  of  the  specifications,  therefore,  requires  more 
rivets  than  are  needed  except  at  the  end  of  the  span.  It 
is  always  necessary,  however,  to  introduce  more  rivets 
near  the  centre  of  span  than  is  required  by  actual  computa- 
tions, for  the  general  stiffness  of  the  girder.  Indeed  even 
more  rivets  are  generally  provided  than  those  prescribed 
in  Art.  3  1  of  the  specifications. 

If  d  is  the  effective  depth  of  the  girder  at  the  end  of 
the  span  and  if  the  end  shear  or  reaction  is  R,  and  if  tA  is 
the  flange  stress  at  the  distance  d  from  the  end  of  span, 
then  will  the  following  equation  of  moments  be  found, 
neglecting  the  negative  moment  of  any  load  within  the 
distance  d  from  the  end  of  the  span: 

Rd  =  tAd. 
Hence 


This  shows  that  an  amount  of  stress  equal  to  the  end 
shear  must  be  given  to  each  flange  within  the  distance  d 
from  the  end.  The  number  of  rivets  required  by  this 
computation  is  a  little  more  than  necessary  if  any  load 
rests  upon  the  girder  between  the  end  and  the  section  at 
the  distance  d  from  it..  It  will  be  clear  that  the  general 
provision  of  Art.  31,  quoted  above,  is  based  upon  this 
end  shear  requirement,  and  it  is  analytically  incorrect, 
but  the  excess  of  rivets  which  it  calls  for  adds  to  the  general 
stiffness  and  capacity  of  the  girder. 

The  weight  of  one  driving  wheel  is  30,000  pounds,  and 
it  is  to  be  distributed  over  three  ties  or  42  inches.     As 


700  PLATE   GIRDERS.  [Ch.  XV. 

the  prescribed  impact  is   100  per  cent.,  the  vertical  load 
per  horizontal  inch  of  girder  will  be: 


42 

It  is  obvious  that  the  flange  stress  taken  by  one-eighth 
of  the  sectional  area  of  the  web  is  received  directly  by  the 
latter  and  does  not  affect  the  rivets  through  the  vertical 
legs  of  the  flange  angles.  If  A\  is  the  actual  net  flange 
section  of  cover-plates  and  angles  and  A  2  the  total  flange 
area,  including  one-eighth  of  the  web  section,  and  if  5 
is  the  total  shear  at  any  moment  section,  while  d  is  the 
effective  depth  of  the  girder,  then  the  horizontal  flange 
stress  H  to  be  taken  up  per  linear  inch  by  the  rivets 

5  A\ 
between  two  sections  the  distance  d  apart  will  be  H  =——. 

Ct  ./i  2 

The  values  of  A\  and  A2,  beginning  at  the  end  section 
of  the  girder,  are  as  follows : 

Section  A ,  A  ^ 

End  2 2. 88  sq.ins.  2 7. 2 6  sq.ins. 

5ft.  22.88  "  27.26  " 

10   "  22.88  "  27.26  " 

15    "  31.88  "  36.26 

25    "  40.88  "  45.26  " 

Centre  40.88  "  45.26  " 

The  unit  (inch)  increments  H  of  horizontal  flange 
stress  found  for  the  various  sections  by  the  preceding 
formula  are: 

T^   j        TT     319*500 VX22. 88  -, 

End       #  =  V   ,     X-     T=333olbs. 


Art.   i io.J  THE  DESIGN  OF  A   PLATE  GIRDER.  701 

15  ft.     H=  =2170  Ibs. 

25    "       #=  =I560     " 

Centre  H  —  =   954    lt 

Each  of  the  above  results  gives  the  horizontal  stress  H 
in  pounds  per  linear  inch,  over  each  80.5  inches  of  girder 
flange  for  each  moment  section  and  to  be  taken  up  by  the 
rivets. 

The  rivet  pitch  p  at  any  section  will  then  be  determined 
by  the  following  formula  if  K  is  the  working  value  of  one 
rivet  in  shear  or  bearing : 


Each  rivet  bears  against  the  web  plate  as  well  as  against 
each  vertical  leg  of  the  flange  angle,  and  as  the  web  plate 
is  much  thinner  than  the  sum  of  the  thickness  of  the  two 
angle  legs,  the  bearing  value  against  the  web  plate  will- 
be  much  less  than  that  against  the  angle  legs.  Furthermore 
each  rivet  is  subjected  to  double  shear,  the  two  shearing 
sections  of  the  rivets  coinciding  with  the  two  faces  of  the 
web  plate.  K,  therefore,  must  be  taken  as  the  least  of  the 
double  shearing  value  and  the  bearing  value  against  the 
web  plate.  The  rivets  to  be  used  are  f-inch  diameter 
before  being  driven  and  the  bearing  value  of  such  a  rivet 
against  a  ^-inch  plate  at  24,000  pounds  per  square  inch 
is  9190  pounds  and  14,430  pounds  in  double  shear  at  12,000 
pounds  per  square  inch,  both  of  these  working  stresses 
being  in  accord  with  the  specifications. 

Applying  the  numerical  results  thus  established  to  the 
formula  for  the  pitch, 


there  will  result: 


702  PLATE   GIRDERS.  [Ch.  XV. 

.,         .,  QIQO 

At  end          p=—=^  y  =2.55  ins. 


5  ft.  point  P  =  -  =2.83 


10         "        f=  =3.18    " 

"  " 


15  P=  =3-53 

25  P=  =4.34 

Centre          p=  =6.26 


If  desired  a  curve  can  be  drawn  at  the  various  points 
with  the  corresponding  pitch  as  a  vertical  ordinate  at  each 
point.  Such  a  curve  will  give  the  rivet  pitch  at  any  point 
in  the  span,  but  such  detail  is  not  usually  required.  The 
above  values  of  the  pitch  may  be  used,  with  judgment, 
without  further  computations  for  any  part  of  the  girder. 
Fig.  i  shows  the  pitch  used  at  the  different  girder  points; 
it  is  frequently  adjusted  to  the  position  of  the  intermediate 
stiff  eners. 


Pitch  of  Rivets  in  Cover- 

The  number  of  rivets  required  in  a  cover-plate  is  at  once 
determined  from  its  net  section.  In  the  present  case  the 
net  section  of  each  cover-plate  is  ^  square  inches,  which, 
at  16,000  pounds,  gives  144,000  pounds  as  the  stress  value 
of  the  plate.  The  rivets  in  the  cover-plates  are  subjected 
to  single  shear  and  the  single-shear  value  of  one  f-inch 
rivet  is  7220  pounds.  Hence  the  number  of  rivets  required 

to  develop  the  full  value  of  one  cover-plate  is  144>ooo  =  2Q 


7220 

rivets.  Between  the  end  of  the  cover-plate,  therefore, 
and  the  point  at  which  the  next  cover-plate  outside  of  it 
begins,  there  must  be  at  least  20  rivets.  As  a  matter  of 
fact  considerably  more  than  that  number  will  be  found, 


Art.   1 10.]  THE  DESIGN  OF  A   PLATE  GIRDER.  703 

as  the  pitch  must  not  exceed  6  inches  in  any  case  and  it 
should  not  be  more  than  3  inches  for  a  distance  of  12  to 
1 8  inches  from  the  end  of  the  plate.  It  will  be  seen  upon 
examining  the  drawing  that  these  conditions  are  fulfilled. 

Top  Flange. 

As  this  flange  is  in  compression,  gross  areas  may  be 
used.  If  the  provisions  of  Art.  30  and  other  Articles  of 
the  specifications  be  scrutinized,  it  will  be  found  that  they 
are  fulfilled  by  the  compression  flange  made  up  as  shown 
in  the  figures,  and  they  need  no  further  detailed  attention. 

End  Stiff eners. 

The  end  stiffeners  must  be  heavy  members  of  their 
class  and  rigidly  riveted  to  the  girder,  as  they  take  the  severe 
impact  or  pounding  at  the  points  of  support  due  to  rapidly 
moving  heavy  locomotives  and  trains.  Art.  79  of  the 
specifications  provides  that  "  There  shall  be  web  stiffeners 
generally  in  pairs,  over  bearings,  at  points  of  concentrated 
loading,  and  at  other  points  where  the  thickness  of  the  web 
is  less  than  one-sixtieth  of  the  unsupported  distance  between 
flange  angles.  .  .  .  The  stiffeners  at  the  ends  and  at  points  of 
concentrated  loads  shall  be  proportioned  by  the  formula 
of  paragraph  16,  the  effective  length  being  assumed  as 
one-half  the  depth  of  girders.  ..."  This  provision 
makes  it  necessary  to  treat  the  end  stiffeners  as  a  column, 
the  working  stress  to  be: 

p  =  16,000  —  70—. 

The  column  load  in  this  case  is  the  maximum  end  shear 
including  impact  allowance  as  given  by  Table  II,  i.e., 
319,000  pounds. 


704  PLATE  GIRDERS.  [Ch.  XV. 


If  two  pairs  of  5  X3^  Xii-inch  angles  be  assumed  for 
trial  with  the  3j-inch  legs  against  the  web  plate,  remem- 
bering that  they  will  be  separated  by  the  thickness  of  the 
plate,  the  radius  of  gyration  of  their  combined  section 
about  an  axis  lying  in  the  centre  of  a  horizontal  web  section 
and  parallel  to  the  web  will  be  3.13  inches.  The  length 

of  the  column  is  -—=40.25  inches  =/.     Hence  the   pre- 

scribed formula  will  give  a  working  stress  of  15,100  pounds 
per  square  inch.  On  this  basis 

A  .     ,     319,000 

Area  required  =  —    —  =  21  sq.ms. 
15,000 

The  actual  sectional  area  of  four  of  the  assumed  angles 
will  be  23.24  square  inches,  which  is  sufficiently  close  to 
the  area  required  to  be  accepted  as  satisfactory. 

The  entire  load  is  carried  to  the  end  stiff  eners  by  the 
|  -inch  rivets  which  bind  them  to  the  web  plate.  The  rivets 
are  in  double  shear  and  bear  on  the  web  plate.  It  has 
already  been  seen  that  the  bearing  value  on  the  web  plate, 
9190  pounds  per  rivet,  is  much  less  than  the  double  shear 

value.     Hence  the  number  of  rivets  required  is  3I9>000       - 

9190 

rivets.  This  computed  number  of  rivets  distributed 
throughout  the  length  of  the  3^-inch  angle  legs  would  make 
the  pitch  too  great.  The  pitch  should  not  exceed  about 
4  inches,  which  would  make  the  number  of  rivets  about 
40.  It  is  essential,  as  already  indicated,  that  the  end 
stiff  eners  be  made  exceptionally  stiff  and  rigid. 

End  stiff  eners  are  not  bent,  but  are  riveted  onto  filling 
plates  having  the  same  thickness  as  the  flange  angle  legs. 
These  filling  plates  enhance  the  stiffness  and  resisting 
capacity  of  the  end  stiff  eners  as  they,  in  fact,  form  a  part 
of  the  latter. 


Art.  I io.]  THE  DESIGN   OF  A  PLATE   GIRDER.  705 

Intermediate  Stiff eners. 

By  referring  to  Art.  79  of  the  specifications  there  will 
be  found  an  empirical  formula  giving  the  maximum  dis- 
tance between  intermediate  stiffeners,  providing,  however, 
that  that  distance  in  no  case  shall  exceed  the  clear  depth  of 
the  web.  Intermediate  stiffeners  are  sometimes  regarded 
as  being  equivalent  to  the  vertical  compression  members 
of  a  Pratt  truss,  but  as  a  matter  of  fact  there  is  no  rational 
system  of  basing  their  design  on  computations.  They 
are  almost  invariably  made  of  angles,  but  sectional  areas 
are  determined  by  experience.  Inasmuch  as  the  total 
transverse  shear  at  the  centre  of  span  is  small,  they  are 
sometimes  omitted  there.  As  a  rule  they  are  never  placed 
farther  apart  than  the  depth  of  web  plate. 

As  this  girder  is  to  carry  a  heavy  railroad  load  pre- 
sumably at  high  speed,  5X3^Xf-inch  steel  angles  will 
be  used  with  the  3!  inch  leg  placed  against  the  web 
plate.  As  the  transverse  shear  increases  toward  the  end 
of  the  span,  the  distance  apart  of  these  intermediate 
stiffeners  will  correspondingly  be  decreased.  In  the  central 
part  of  the  span  this  distance  is  seen  to  be  5  feet  if  inches, 
but  near  the  ends  it  is  reduced  to  3  feet  5!  inches.  The 
pitch  of  the  rivets  in  these  intermediate  stiffeners  may  vary 
from  3  inches  to  5  or  6  inches,  the  greater  pitch  being  near 
the  mid  depth  of  the  web. 

Splices  in  Flanges. 

It  will  be  found  that  cover-plates  and  flange  angles 
may  be  purchased  of  full  lengths  required  on  this  plate 
girder.  When,  in  general,  the  girders  are  so  long  as  to 
require  splicing  of  the  parts  of  flanges,  those  joints  for  the 
tension  flange  must  be  so  designed  as  to  leave  the  net 
section  as  large  as  practicable,  as  the  entire  stress  must  be 


706  PLATE  GIRDERS.  [Ch.  XV. 

carried  by  the  net  section.  It  is  good  practice  and  cus- 
tomary not  to  have  two  joints  in  adjacent  parts  concur, 
i.e.,  there  should  be  breaking  of  joints  so  as  to  have  a 
joint  in  one  part  only  of  the  flange  at  the  same  section. 
In  this  manner  the  net  section  at  each  joint  may  attain 
its  maximum  value.  In  the  splicing  of  angles  both  legs 
should  be  spliced.  In  compression,  riveted  joints  can 
scarcely  be  expected  to  transfer  stresses  by  abutting  sur- 
faces in  those  joints.  They  should  be  spliced  about  as 
effectively  as  tension  joints,  although  the  question  of  net 
section  does  not  arise,  the  gross  section  being  available. 

Splices  in  Web  Plates. 

As  one-eighth  of  the  gross  web-plate  section  is  considered 
as  resisting  bending  as  a  part  of  the  flange  area,  the  rivets 
at  a  web-plate  splice  must  be  sufficient  to  resist  the  cor- 
responding bending  moment.  This  web-plate  moment  is, 
therefore, 


=  5,670,000  in.-lbs. 
8  io 

There  must  be  two  splice-plates,  one  on  each  side  of 
the  web,  each  of  which  need  not  be  as  thick  as  the  main 
plate,  but  in  this  case  f-inch  splice-plates  have  been  used 
so  that  the  intermediate  stiff  ener  need  not  be  bent.  For 
this  size  of  girder  there  should  be  three  rows  of  rivets  on 
each  side  of  the  joints.  If  it  be  assumed  that  the  pitch 
be  A  inches  in  each  row,  there  will  be  nine  rivets  in  each  of 
the  three  rows  between  the  mid  depth  of  the  web  and  the 
back  of  the  flange  angles.  If  the  loads  carried  by  these 
rivets  in  resisting  bending  vary  directly  as  the  distance 
from  the  neutral  axis  at  mid  depth,  their  resultant  will 
act  at  1X40  =  26.7  inches  from  that  line.  The  bearing 


Art.  iio.l  THE  DESIGN  OF  A  PLATE  GIRDER.  707 

value  of  a  {-inch  rivet  against  the  ^-inch  web  is  9190 
pounds.  Hence  the  resisting  moment  of  the  54  rivets 
on  one  side  of  the  joint  is: 


.  7  =6,600,000  in.-lbs. 


As  this  is  greater  than  5,670,000  in.-lbs.,  the  proposed 
arrangement  of  the  joint  is  satisfactory.  The  two  splice- 
plates  will,  therefore,  each  be  19  Xf  inches  by  5  feet  8|  inches, 
as  shown  in  Fig.  i. 

In  general  every  joint  splicing  should  be  tested  for 
the  transverse  shear  which  it  must  carry.  In  this  instance 
it  is  clear  that  the  splice-plates  will  carry  more  shear  than 
the  web. 

General  Considerations. 

The  girder  proper  with  its  flanges,  web,  and  stifleners 
has  been  designed  in  this  article  without  indicating  the 
manner  of  connecting  such  lateral  or  cross  bracing  as 
would  be  required  in  the  complete  design  of  a  railroad 
plate-girder  span.  The  design  of  such  bracing  would  be 
supplementary  to  the  actual  design  of  the  girder  as  made, 
and  it  is  the  purpose  here  to  illustrate  only  those  principles 
belonging  to  the  design  of  the  girder  proper.  The  design 
of  the  bracing  and  the  details  of  its  connection  with  the 
girder  belong  rather  to  bridge  construction  than  to  the 
subject  treated  here.  Fig.  2  has  been  introduced,  however, 
as  an  illustration  to  indicate  the  general  features  of  the 
complete  structure. 

Large  plate  girders  are  not  always  built  complete  in 
the  shop,  although  girders  nearly  100  feet  in  length  are 
frequently  and  perhaps  usually  so  completed  at  the  present 
time.  When  it  is  necessary  to  build  them  in  portions 


708  PLATE  GIRDERS.  [Ch.  XV. 

and  rivet  the  portions  together  in  the  field,  the  general 
principles  governing  the  construction  of  the  necessary 
field-joints  are  precisely  the  same  as  those  illustrated  in 
this  article.  They  are  simply  adjusted  or  adapted  to  the 
exigencies  of  each  particular  case. 

The  bill  of  material  and  estimated  weight  of  a  single 
girder  as  designed  is  as  follows  : 

Pounds. 
Two  80"  XT*"  web  plates,  21'  n£"  long  .............     5,236 

One  80"  XiV'  web  plate,  26'  \"  long  .................     3,094 

Four  6"X6"Xf"  angles,  70'  long  ...................     8,036 

One  14"  Xl"  cover-plate,  70'  long  ....................     2,499 

One  14"  X  I"  cover-plate,  55'  5^"  long  ................ 

Two  14"  X|"  cover-plates,  47'  6£"  long  .............. 

Two  14"  X?"  cover-plates,  33'  3"  long  ............... 

Eight  5"X3i"XH"  ar-gles,  6'  7"  long  ............... 

Twenty-eight  5"X3l"Xf"  angles,  6'  7"  long..  ........ 

Four  io"Xf"  filler-plates,  5'  8|"  long  ................ 

Four  19"  Xf"  splice-plates,  5'  8|"  long  ............... 

Twenty-four  3l"Xf"  filler-plates,  5'  8|"  long  ......... 

Two  14"  Xf"  sole-plates,  i'  6"  long  ..................        107 

Rivets  ........................  .  ..................          800 

Total  for  one  girder  .....  .......................   30,502 

The  weight  of  girder  per  linear  foot  therefore  is: 


70 

If  the  plate  girder  were  of  the  through  type,  there 
would  be  no  change  whatever  in  the  procedures  of  design 
which  have  been  followed,  but  in  order  to  give  a  better 
appearance  to  the  ends  they  would  be  formed  as  shown  in 
Fig.  5-  The  latter  figure  shows  the  same  end  stiffness, 
depth  of  girder  and  the  same  flange  angles  as  Fig.  i. 

Art.  in.  —  Length  of  Cover-plates. 

There  are  various  methods  of  determining  the  lengths 
of  cover-plates  of  plate  girders  involving  simple  compu- 


Art.   in.]  LENGTH   OF  COVER-PLATES.  709 

tations  only,  which  are  well  illustrated  by  the  following 
procedures  : 

The  first  of  these  procedures  is  based  on  the  assump- 
tion that  the  depth  of  the  girder  is  uniform  and  that  the 
bending  moment  throughout  the  length  of  girder  varies 
as  the  ordinate  of  a  parabola  as  in  the  case  of  uniform 
loading.  The  following  notation  is  required: 

/  =  effective  length  of  span  either  in  feet  or  inches  ; 
L==  length  of  cover-plate  required  in  the  same  unit  as  /; 
A  =  total  net  flange  area  ; 
a  =  net  cover-plate  area  required. 

Since  the  flange  and  cover-plate  areas  vary  directly 
as  the  flange  stresses,  and  as  the  latter  vary  as  the  ordi- 
nates  of  a  parabola  when  the  depth  of  girder  is  constant, 
the  following  equation  will  result: 


_ 
I2  ~  A* 

or 


(0 


Eq.  (i)  will  give  the  length  of  the  cover-plate  whose 
area  of  section  is  a.  Any  convenient  unit  may  be  taken 
for  a  and  A,  but  the  square  inch  is  ordinarily  employed. 

If  there  are  several  cover-plates,  a  is  to  be  taken  suc- 
cessively the  area  of  the  first,  second,  third,  etc.,  cover- 
plates  in  summation,  i.e.,  it  will  first  be  taken  as  the  net 
sectional  area  of  the  top  cover,  then  as  the  net  sectional 
area  of  the  top  cover  added  to  that  of  the  cover-plate 
below  it,  and  so  on. 

The  second  method  is  the  following,  and  is  applicable 


710  PLATE   GIRDERS.  [Ch.  XV. 

to  the  case  of   a'  girder  with  varying  depth,  the  notation 
being  as  follows : 

Let  w  =  uniform  load  per  linear  foot,    or   "  equivalent 

uniform  load"  per  linear  foot; 

d  and  d'  represent  the  effective  depths  of  girder  in 
feet  at  the  centre  of  span  and  at  the  end  of 
the  cover-plate  respectively; 
A  —  a  =  a'  =  area   of   flange    section   at   the    end    of 

cover-plate ; 
T  =  permissible  flange  stress  per  square  inch; 

the  bending  moment  at  the  end  of  the  cover-plate  will 
then  be 

/2     w/L\  U 

M  =  w^ =  AdT-w-^-  =  d'a'T.     .     .     (2) 

Q         2   \2  J  O 

By  solving  the  second  and  third  members  of  the  pre- 
ceding equation  there  will  result 


-    \(Ad-a'd')T  _^          (Ad-a'd')T 

It  must  be  remembered  that  the  application  of  either 
of  the  two  preceding  methods  will  give  the  net  length  of 
the  cover-plate.  There  must  be  added  12  to  18  ins.  at 
each  end  with  rivets  closely  pitched  so  that  the  cover- 
plate  may  certainly  take  its  stress  at  the  points  where  its 
effectiveness  should  begin. 

Art.  112. — Pitch  of  Rivets. 

A  simple  method  of  finding  the  pitch  of  rivets  piercing 
the  vertical  legs  of  the  flange  angles  and  the  web  plate  of  a 


Art.   112.]  PITCH   OF  RIYETS.  711 

plate  girder  at  any  section  of  the  beam  may  readily  be  found 
by  using  the  general  but  elementary  expression  for  the  bend- 
ing moment, 


By  differentiating  this  equation,    . 
2P.doc  =  Sdx=dM\ 

S  representing  the  total  transverse  shear. 

If  dM  is  the  change  of  bending  moment  for  the  distance 
along  the  flange  represented  by  the  pitch  of  rivets,  p,  the 
change  of  flange  stress  for  the  same  distance  will  be  found 
by  dividing  dM  by  the  effective  depth  of  the  girder,  d.  If 
the  pitch  of  rivets,  p,  be  placed  in  the  preceding  equation  in 
place  of  dx,  the  corresponding  change  of  flange  stress  will 
represent  the  amount  of  stress  transferred  to  the  flange  by 
one  rivet.  Representing  that  variation  of  flange  stress  by 
v,  the  last  of  the  preceding  equations  may  be  written 


In  this  equation  v  represents  either  the  bearing  capacity 
of  one  rivet  against  the  web  plate  or  against  the  two  flange 
angles,  or  the  double  shearing  value  of  the  same  rivet,  i.e., 
the  least  of  those  three  values.  Ordinarily  the  bearing  of 
the  rivet  against  the  web  plate  will  be  less  than  either  of  the 
two  other  quantities;  hence  that  bearing  value  would  then 
be  substituted  for  v.  In  general  the  least  of  the  three  pre- 
ceding values  for  one  rivet  is  to  be  substituted  for  v  in  an 
actual  computation.  The  total  transverse  shear  S  is  always 
known  at  any  section  or  may  readily  be  determined.  The 
preceding  formula  for  the  pitch,  therefore,  is  a  very  simple 
one  and  is  much  employed. 


CHAPTER  XVI. 

MISCELLANEOUS  SUBJECTS. 

Art.  113. — Curved  Beams  in  Flexure. 

IF  beams  are  sharply  curved,  i.e.,  if  the  radius  of  curva- 
ture of  the  neutral  surface  is  comparatively  small,  the  for- 
mulae expressing  the  common  theory  of  flexure  for  such 
beams  will  contain  the  radius  of  curvature  and  corre- 
sponding variations  from  the  formulae  for  straight  beams. 

Let  Fig.  i  represent  part  of  a  curved  beam  subjected  to 
flexure,  AC  representing  the  radius  of  curvature  at  the 


point  A  before  flexure  while  C'A'  represents  the  radius 
of  curvature  of  the  same  surface  after  flexure  takes  place. 
OAO'  represents  the  neutral  surface.  A'b"  is  the  continu- 
ation of  C'A'.  Similarly  A'b  is  the  continuation  of  CA'. 
Finally,  A'b'  is  drawn  parallel  to  CA.  def  represents  the 
normal  section  of  the  beam  and  A  A'  is  supposed  to  be 
a  differential  of  the  length  of  the  neutral  surface. 

712 


Art.   113.]  CURVED  BEAMS  IN  FLEXURE.  713 

The  ordinate  ±y  is  measured  from  A  as  an  origin 
toward  B  or  D,  respectively,  z  is  the  varying  width  of 
the  normal  section  of  the  beam  and  hence  it  is  measured 
normal  to  y  and  x,  the  latter  being  measured  along  OAO'. 
A  differential  of  the  section  of  the  beam  is  zdy. 

As  the  normal  sections  of  the  beam  are  assumed  to 
remain  plane  after  flexure,  let  the  rate  of  strain,  i.e.,  the 
strain  per  unit  of  length  of  fibre  at  any  point  distant  y 
from  the  neutral  surface  be  uy,  u  being  the  apparent 
rate  of  strain  at  unit  distance  from  the  neutral  surface. 

By  referring  to  Fig.  i  there  may  at  once  be  written: 


b'b  =dx\    bb"  = 
By  similarity  of  triangles, 


y 

This  equation  gives  at  once : 


r-r'        r' 


r 
—  i 


-  -- 


If  the  beam  were  originally  straight,  in  which  case  the 
radius   of   curvature   r  =  co  ,  eq.    (2)   would   take  the   form 

u=  —  ,  the  usual  expression  for  the  rate  of  strain  at   unit 

distance  from  the  neutral  surface  of  a  straight  beam. 
If  again  the  radius  of  curvature  is  sufficiently  large,  so  that 
r  may  be  written  for  r+y  without  sensible  error: 


(3) 


714  MISCELLANEOUS  SUBJECTS.  [Ch.  XVI. 

This  expression  for  u  may  be  used  for  curved  beams  if 
the  curvature  is  not  too  sharp. 

If  the  radius  r  is  infinitely  great,  u  =  -f,  which  is  the 

value  for  a  straight  beam. 

Eq.  (2)  shows  that  the  rate  of  strain  u  at  unit  distance 
from  the  neutral  surface  and  corresponding  to  the  rate  of 
strain  at  any  distance  y  is  variable,  as  y  appears  in  the 
denominator  in  such  a  way  as  to  make  u  smaller  the 
greater  the  distance  of  the  fibre  from  the  neutral  surface. 
This  is  in  consequence  of  the  curvature  of  the  beam  and 
results  from  the  assumption  that  normal  sections  plane 
before  flexure  remain  plane  after  flexure.  With  the 
increase  of  length  of  fibre  due  to  curvature  as  its  distance 
from  the  neutral  axis  increases,  a  less  rate  of  strain  is 
required  to  keep  the  section  plane  after  flexure.  This 
assumption  is  not  strictly  true,  and  it  may  be  a  matter  of 
doubt  whether  it  is  necessary  or  advisable  even  in  the 
interests  of  correct  analysis. 

If  k  is  the  fibre  stress  of  tension  or  compression  at  any 
distance  y  from  the  neutral  axis,  there  may  be  at  once 
written  : 


The  stress  on  an  element  zdy  of  the  section  will  then  be  : 

..    .    fa, 


Let  kf  and  k"  be  the  intensities  of  stress  at  the  distances 
y'  and  —  y"  from  the  neutral  surface.     Then  by  eq.  (4)  : 

*'        /    r-y" 


k"     r+y'   -y1 


Art.  113.]  CURBED  BEAMS  IN  FLEXURE.  715 

From  this  equation  : 


v>  —v-,,  ....  (Sfl) 

r-y" 


If  y'=y"t  eq.  (50)  becomes: 


Eq.  (5&)  shows  that  the  intensity  of  stress  at  a  given 
distance  from  the  neutral  axis  will  be  greater  on  the  concave 
side  of  the  curve  than  on  the  convex,  and  that  this  relation 
holds  until  the  radius  of  curvature  becomes  infinitely 
great. 

In  order  to  locate  the  neutral  axis  the  integral  of  the 
two  members  of  eq.  (5)  between  the  limits  of  y  and  —y 
must  be  placed  equal  to  zero,  giving  eq.  (6)  : 


Again,  the  bending  moment  formed  by  the  direct 
stresses  of  tension  and  compression  in  the  section  may  be 
written  in  the  usual  manner  as  follows,  M  representing  the 
moment  : 


Eq.  (6)  shows  that  the  neutral  axis  will  not  pass  through 
the  centre  of  gravity  of  the  section.  As  the  intensity  of 
stress  on  the  convex  side  of  the  curve  will  be  less  than  if 
the  beam  were  straight,  the  neutral  axis  will  be  on  that 


7i6 


MISCELLANEOUS  SUBJECTS. 


[Ch.  XVI. 


side  of  the  centre  of  gravity  of  the  section  toward  the  con- 
cave surface  of  the  beam.  Eq.  (7)  shows,  again,  that  the 
integral  is  not  the  moment  of  inertia  of  the  section  about 
the  neutral  axis,  but  it  will  reduce  to  that  if  the  radius 
of  curvature  r  be  supposed  infinitely  great. 

The  integrations  shown  in  the  second  members  of  eqs. 
(6)  and  (7)  can  at  once  be  made  when  the  form  of  cross- 
section  is  known.  Inasmuch  as  this  analysis  for  curved 
beams  finds  one  of  its  important  applications  in  connection 
with  the  design  and  carrying  capacity  of  large  hooks,  a  trape- 
zoidal cross-section  shown  in  Fig.  2  will  be  assumed  by 

way  of  illustration,  and  from 
that  the  rectangle  section  at 
once  results.  In  that  figure 
the  larger  end  CD  of  the  trap- 
ezoid  will  be  considered  to 
lie  in  the  concave  or  inner 
surface  of  the  hook  and  at 
right  angles  to  the  plane  of 
the  hook.  As  the  trapezoid 
is  symmetrical,  a=%FH,  and 

i the    angle  of  inclination  of  a 

'FIG.  2.  sloping   side   as    HD    to    the 

centre  line  will  be  taken  as  a, 

Then  z  will  represent  one-half  of  the  width  of  the  trapezoid 
at  any  point: 


—y)  tan  a. 


(8) 


If  z  be  inserted  in  eqs.  (6)  and  (7)  there  will  be  required 
the  following  integrations  in  which  yi+yo=d: 


f 

J- 


r-y0 


(9) 


Art.  113.]  CURBED  BEAMS  IN  FLEXURE.  717 

r*  log  '-+£,  (I0) 

& 


•  •  •  <"' 


If  these  values  of  z  and  the  integrals  given  in  eqs.  (9), 
(10)  and  (n)  be  substituted  in  eq.  (6),  there  will  at  once 
result  : 


d r+-  tana+a 


As    known    quantities    let    r+y\=R    and    r—  yo=Ro, 
then  eq.  (12)  may  take  the  form: 

7?  /d  \ 

r  log  -^-(R  tan  a  -fa)  =  r  d  tan  a  +d(  -  tan  a  -fa  1 . 
KO  \2  / 

Hence : 

f-I-  R  v  — ^-       •       •       (13) 

(7?  l°g"B — ^)  ^an  <*+#  log  — 


After  r  is  determined  by  eq.   (13)  there  will  at  once 
result : 

yi=R-r     and     yo=d—yi.       .     .     .     (14) 

If  the  section  is  rectangular,  «=tan  «=o,  hence, 

'  = B-     and    yi=R--d—-.     m     . 

logf  logj 

*M>  /M) 


718  MISCELLANEOUS  SUBJECTS.  [Ch.  XVI. 

If  the  section  is  triangular,  a  =  o  and  the  second  member 
of  eq.  (13)  will  be  correspondingly  simplified  as  follows: 


As  this  expression  is  independent  of  a,  y\  and  yQ  remain 
unchanged  whatever  may  be  the  value  of  that  angle. 

Having  thus  found  y\  and  y0,  the  position  of  the  neutral 
axis  of  the  section  is  determined  and  the  expression  for 
the  bending  moment  can  now  be  written  by  the  aid  of 
eqs.  (4)  and  (7),  the  latter  being  the  general  expression 
for  the  bending  moment.  By  the  aid  of  eq.  (4)  the  inr 
tensity  of  stress  in  the  extreme  fibre  at  the  distance  yQ 
from  the  neutral  axis  may  be  written  as  follows  : 


-.    ,  (16) 

-yo 

Hence, 

......  <•» 

By  introducing  the  second  member  of  eq.  (17)  in  eq. 
(7)  as  well  as  the  value  of  z  from  eq.  (8)  and  the  integrals 
given  in  eqs.  (10)  and  (n),  the  following  value  of  the 
moment  M  will  result  : 


2k0(r-yo)  Cyi    /  \ 

(18) 


yo 


-y0  r+y] 

,  d 

tan 


j        91      f~yi  ana/J9  N  ,     ^ 

~cfr+r2log-—^-j-     —  —  (dt-wiyon-         .     -     (19) 


Art.  113.]  CURVED  BEAMS  IN  FLEXURE.  719 

As  is  evident,  the  factor  2  appears  in  the  second  members 
of  eqs.  (18)  and  (19),  for  the  reason  that  the  section  taken 
is  symmetrical  and  the  varying  ordinate  z  is  half  the  width 
of  section  at  any  point.  If  a  were  taken  as  the  extreme 
width  of  section  on  the  narrow  side  instead  of  half  that 
width  and  if  a  were  to  be  so  taken  that  (y\—y)  tan  a. 
added  to  a  represents  the  full  width  of  the  section  at  the 
point  located  by  y,  the  factor  2  would  be  omitted  from  the 
second  member  of  the  value  for  M. 

If  the  section  is  rectangular  a  =  tan  a  =  o  and  the 
expression  for  the  moment  M  then  becomes  : 


(2o) 


If  the  section  were  triangular  a  =  o  in  the  second  member 
of  eq.  (19). 

These  equations  may  be  employed  in  the  design  of  curved 
beams  of  any  form  of  cross-section  or  degree  of  curvature 
when  those  based  on  the  common  theory  of  flexure  for 
straight  beams  are  not  applicable.  As  a  general  statement 
it  may  be  said  that  the  formulae  for  straight  beams  may  be 
used  without  essential  error  in  all  cases  except  those  of  such 
special  character  as  hooks  and  other  structural  or  machine 
members  in  which  the  curvature  is  sharp.  The  applica- 
tion of  the  preceding  formulae  to  the  case  of  hooks  will  be 
illustrated  in  the  next  article. 

Art.  114.  —  Stresses  in  Hooks. 

The  diagram  of  a  hook  shown  in  Fig.  i  illustrates  the 
conditions  of  loading  to  which  hooks  in  general  are  sub- 
jected. The  material  to  the  right  of  the  point  of  applica- 
tion of  the  load  is  subjected  to  no  stress  whatever  except 
in  a  secondary  way  near  that  point.  On  the  left  of  the 


720 


MISCELLANEOUS   SUBJECTS. 


[Ch.  XVI. 


load,  however,  the  arc  of  the  hook,  supposed  to  be  circular 
in  this  case,  is  subjected  to  direct  stress,  shear  and  .bending, 
the  bending  moment  increasing  as  that  part  of  the  hook 


parallel  to  the  loading  is  approached,  but  it  decreases  in 
passing  on  to  the  shaft  of  the  hook  supposed  to  be  in  line 
with  the  load.  The  section  of  maximum  bending  AB 
is  subjected  to  the  combined  direct  pull  of  the  load  and 


Art.   114.] 


STRESSES   IN  HOOKS. 


721 


the  bending  moment  equal  to  the  load  multiplied  by  the 
normal  distance  from  its  line  of  action  to  the  centre  of 
gravity  of  the  section.  This  cross-section  of  greatest 
bending  moment  will  first  be  treated  as  if  subjected  to 
pure  flexure.  The  necessary  simple  analysis  required  to 
determine  the  greatest  intensity  of  stress  in  the  section  will 
then  be  made.  In  the  section 
of  greatest  bending  moment 
there  is  no  shear. 

The  cross-section  of  the 
main  part  of  a  hook  maybe 
taken  as  approximately  trap- 
ezoidal, as  shown  in  Figs,  i 
and  2.  In  the  present  in- 
stance the  greatest  dimension 
of  this  cross-section  lying  in 
the  central  plane  of  the  hook 
will  be  taken  as  5  inches  and 
the  corners  will  be  rounded 
approximately  as  shown. 

Obviously  the  integrations 
of  eqs.  (9),  (10)  and  (n)  of  the 
preceding  article  do  not  rep- 
resent accurately  the  approx- 
imate trapezoid  of  Fig.  2 .  This 
integration  or  its  equivalent, 

however,  may  be  accomplished  with  sufficient  accuracy 
by  a  number  of  approximate  processes,  i.e.,  by  transformed 
figures  and  by  dividing  the  section  into  a  sufficient 
number  of  small  parts.  A  simpler  method  and  one  giving 
reasonably  accurate  results  is  to  draw  two  lines  F'C'  and 
FD  in  such  a  way  as  to  make  a  true  trapezoid  whose  resist- 
ing moment  will  be  essentially  the  same  as  the  approx- 
imate trapezoid.  This  will  be  accomplished  if  the  two 


FIG.  2. 


722  MISCELLANEOUS  SUBJECTS.  [Ch.  XVI. 

lines  indicated  be  drawn  in  such  a  way  that  each  area 
between  a  broken  line  as  F'C'  and  the  inclined  full  line 
of  the  actual  section  be  three  times  the  combined  area 
between  CB  and  the  curved  end  of  the  section  and  between 
AF'  and  the  other  curved  end  of  the  section.  This  rela- 
tion results  from  the  fact  that  the  bending  stress  between 
the  two  lines  indicated  varies  in  intensity  from  zero  at  the 
neutral  axis  to  nearly  the  maximum  in  the  extreme  fibre 
of  the  section  and  has  its  centre  at  two-thirds  of  the  distance 
from  the  neutral  axis  to  the  extreme  fibre.  The  relation 
indicated,  therefore,  makes  the  bending  moments  of  the  two 
parts  inside  and  outside  of  the  actual  section  equal.  This 
construction  will  give  for  one-half  the  modified  figure: 


.43  inch=a;  BC  =  i.i  inches;  a  =8°  30'; 
tan  a  =.148;  ^  =  5+2.2=7.2  inches;  R  =  2.2  inches. 
d  =  5  inches. 

Fig.  i  shows  that  R  and  RQ  are  the  interior  and  exterior 
radii  respectively  of  the  arc  of  the  hook  where  the  section 
of  greatest  bending  moment  exists.  By  introducing  these 
numerical  quantities  in  eq.  (13)  of  the  preceding  article 
there  will  at  once  result : 

r  =3.87  inches. 

Hence, 

y\=R—  r  =3. 33  inches; 

yo  =d—y\  =1.67  inches. 

By  inserting  the  same  numerical  values  together  with 
y\  and  y0  in  eq.  (19)  of  the  preceding  articles,  the  value  of 
the  bending  moment  becomes : 

M  =4.88^0.  (i) 


Art.   114.]  STRESSES   IN  HOOKS.  723 

This  moment  obviously  can  be  expressed  in  terms  of  the 
intensity  of  stress  in  the  extreme  fibres  on  the  opposite 
side  of  the  section,  i.e.,  3.33  inches  from  the  neutral  surface. 
By  eq.  (4)  of  the  preceding  article: 


yo(r+yi) 

After  substituting  the  values  of  the  quantities  already 
determined  there  will  be  found  ki=.6iko.  Or  there  may 
be  written  from  the  same  eq.  ko  =  i.64.k\.  The  bending 
moment  expressed  in  terms  of  the  greatest  intensity  of 
stress  in  the  extreme  fibres  is  obviously  the  form  desired 
for  practical  purposes. 

Let  the  hook  shown  in  Fig.  i  be  supposed  to  carry 
a  load  of  20,000  pounds.  The  centre  of  gravity  G  of 
the  actual  cross-section  is  2.13  inches  from  the  side  CD 
of  the  cross-section,  Fig.  2.  Hence  the  load  assumed 
will  cause  a  bending  moment  about  the  line  GH  equal 
to  20,  oooX(2.  13  +2.2  =4.33)  =86,600  inch-pounds.  It  is 
to  be  observed  that  inasmuch  as  the  20,000  pounds  is  taken 
as  uniformly  distributed  over  the  cross-section  the  lever 
arm  of  the  load  is  the  normal  distance  from  its  line  of 
action  to  the  centre  of  gravity  of  that  section,  although  the 
resisting  moment  of  internal  stresses  has  the  axis  deter- 
mined by  eq.  (14)  of  the  preceding  article,  the  two  axes 
being  parallel  to  each  other. 

The  greatest  intensity  of  tensile  bending  stress  in  the 
section  therefore  takes  the  following  value: 

7  86,6OO  -M  •  /       \ 

ko  =  —  '—  -  =  17,740  Ibs.  per  sq.in:       .     .     (2) 
4.00 

The  uniformly  distributed  tensile  stress  equal  to  the 
load  will  act  upon  the  entire  actual  area  of  section,  which 


724  MISCELLANEOUS  SUBJECTS.  [Ch.  XVI. 

is   7.9   square  inches.     Hence,   that   tensile  intensity  will 

be  — ! =2530  pounds  per  square  inch.     The  resultant 

7-9 
greatest  intensity  of  stress  in  the  entire  section  will  be: 

17,740  +  2530  =  20,270  Ibs.  per  sq.in.      .     .     (3) 

The  resultant  intensity  on  the  opposite  side  of  the  sec- 
tion at  A,  Fig.  2,  will  be,  since  ki=.6iko\ 

—  17,740  X.6i  +2530  =  —8291  Ibs.  per  sq.in.     .      (4) 

The  minus  sign  is  used  because  the  bending  stress  is 
compression  throughout  that  part  of  the  section  indicated 
by  yi. 

It  is  commonly  observed  in  actual  experience  that 
hooks  or  other  similar  bent  members  break  at  the  inside 
of  the  section  where  the  curvature  is  the  sharpest.  The 
eqs.  (4)  and  (56)  of  the  preceding  article  indicate  clearly 
the  reason  for  such  failures  as  the  intensity  of  stress  k  in 
the  extreme  fibre  is  shown  to  vary  inversely  with  the  radius 
of  curvature  r+y.  When,  therefore,  the  curvature  is 
sharp,  i.e.,  the  radius  of  curvature  is  small,  the  fibre  stress 
k  increases  rapidly,  especially  on  the  inside  of  the  curve 
where  the  radius  of  curvature  is  r—y. 

This  example  shows  the  general  method  of  treating  the 
stresses  in  hooks  by  the  common  theory  of  flexure  based 
on  the  assumption  that  normal  sections  plane  before  flexure 
remain  plane  after  bending. 

It  is  well  known  that  this  assumption  is  not  strictly 
correct,  and  it  is  further  known  that  the  ordinary  or  com- 
mon theory  of  flexure  is  not  accurately  applicable  to  such 
short  beams  as  are  contemplated  in  the  theory  of  hooks. 


Art.  114.]  STRESSES   IN  HOOKS.  725 

Comparison  with  the  Theory  of  Flexure  for  Straight  Beams. 

It  is  indicated  above  that  the  assumptions  on  which  the 
preceding  analyses  are  based  are  not  strictly  correct.  If 
it  be  assumed  that  the  intensity  of  stress  varies  directly 
as  the  distance  from  a  neutral  axis  passing  through  the 
centre  of  gravity  of  the  section,  as  for  straight  beams,  and 
if  k'i  is  the  greatest  intensity  of  stress  in  the  extreme 
fibres  (FFf,  Fig.  2)  the  bending  moment  will  be: 


In  this  equation  I  is  the  moment  of  inertia  about  an  axis 
through  the  centre  of  gravity  G,  Fig.  2,  while  yc  is  jthe 
distance  of  that  axis  from  the  most  remote  fibre  at  A. 
The  moment  of  inertia  /  of  the  actual  section  shown  in 
Fig.  2  about  a  neutral  axis  through  the  centre  of  gravity 
G  at  the  distance  2.87  inches  from  A  is  14.9.  Hence, 
the  bending  moment  on  the  preceding  assumption  is: 


(6) 


2.87 


C    *2 

As    the    fraction    -—^=1.07    this    assumption    is    seen 
4.88 

to  give  a  result  only  7  per  cent,  greater  than  that  of  the 
analysis  for  curved  beams  if  the  extreme  fibre  stress  is  the 
same  in  amount  in  both  cases.  It  is  true  that  the  result 
has  the  apparent  defect  of  placing  the  greatest  intensity 
of  stress  on  the  wrong  end  of  the  section. 

Art.  115.—  Eccentric  Loading. 

The  analysis  of  stresses  produced  in  a  column  or  other 
structural  member  by  eccentric  loading  has  already  been 


726 


MISCELLANEOUS  SUBJECTS. 


[Ch.  XVI. 


discussed  in  preceding  articles,  but  it  is  desirable  to  con- 
sider some  further  and  more  general  features  of  that  analysis. 
A  column  or  structural  member  is  said  to  be  eccentric- 
ally loaded  when  it  carries  a  force  or  load  acting  parallel 
to  its  axis  but  not  along  that  axis.  The  perpendicular 


FIG.  i. 


distance  between  the  axis  of  the  piece  and  the  line  of  action 
of  the  load  is  called  the  eccentricity  of  the  latter. 

Let  Fig.  i  represent  the  normal  cross-section  of  such  a 
member  when  the  load  P  acts  at  any  point  Q  in  that  cross- 
section.  The  load  P  will  then  act  parallel  to  the  axis  of 
the  piece,  but  at  the  distance  CQ  from  it,  C  being  supposed 
to  be  the  centre  of  gravity  of  the  section.  The  ellipse 


Art.  115.]  ECCENTRIC  LOADING.  727 

drawn  with  C  as  its  centre  is  the  ellipse  of  inertia,  the  semi- 
axes  ri  and  r?  being  the  principal  radii  of  gyration  of  the 
normal  section.  Any  semi-diameter  as  CQ'  represents 
a  radius  of  gyration  rr. 

If  the  force  or  load  P  acts  at  any  point  whatever,  as  Q, 
and  parallel  to  the  axis  of  the  piece,  it  will  create  a  bending 
moment  equal  to  PxQC.  If  #'  and  y'  are  the  coordinates 
of  Q  the  components  of  that  moment  will  be  Px'  and  Py' ', 
the  former  about  the  axis  Y,  the  latter  about  the  axis  X. 
1 1  and  1 2  being  the  principal  moments  of  inertia,  as  already 
indicated,  the  intensities  of  bending  stresses  produced  by 
these  two  component  moments  at  any  point,  whose  co- 

Px'  PV 

ordinates  are  x  and  y,  will  be  — — x  and  —^-y,  respectively. 

^1  -L2 

Furthermore,  the  load  P  will  produce  a  uniform  normal 

stress  over  the  entire  cross-section  of  the  member,  if  A 

p 

is  the  area  of  that  cross-section,  represented  by  — .  The 

A 

resultant  intensity  of  stress  k  therefore  at  any  point  of  the 

section  will  be : 

,      P  ,  Px'     ,  Py' 


T)  /  /  /     \ 

"'"   k=A\I+7^+^) (l) 

At  the  neutral  axis  the  intensity  of  stress  is  equal  to  zero, 
hence, 


Eq.  (2)  is  the  equation  of  a  straight  line,  i.e.,  the  neutral 
axis,  along  which  the  intensity  of  stress  is  zero,  x  and  y 
being  the  variable  coordinates.  It  is  obvious  from  eq.  (2) 
in  connection  with  the  general  considerations  respecting 


728  MISCELLANEOUS  SUBJECTS.  [Ch.  XVI. 

the  action  of  the  load  P  that  the  position  of  the  neutral 
axis  will  depend  upon  the  magnitude  of  that  load  and  the 
distance  of  its  line  of  action  from  the  axis  of  the  member. 
If  x  and  y  are  zero,  there  will  be  no  bending,  and  the  section 
of  the  member  will  be  subjected  to  uniform  compression 
only. 

If  the  point  of  application  Q  of  P  is  on  the  curve  its  coor- 
dinates %'  and  y'  must  satisfy  the  equation  of  the  ellipse  : 


The  equation  of  a  straight  line  tangent  to  the  ellipse 
at  a  point  whose  coordinates  are  x'  and  y1  is  : 


When  the  point  of  application  of  P  is  on  the  ellipse,  x' 
and  yf  have  the  same  values  in  eqs.  (2)  and  (4).  Hence  in 

that  case  -^  also  has  the  same  value  in  the  two  equations, 
dx 

showing  that  the  neutral  axis  is  parallel  to  the  tangent  to 
the  ellipse  at  the  point  where  P  acts.  If  in  eq.  (4)  —%' 
and  —  y'  be  substituted  for  +#  and  -\-y,  that  equation  will 
become  identical  with  eq.  (2),  i.e.,  for  this  case  the  neutral 
axis  is  tangent  to  the  ellipse  at  a  point  diametrically  opposite 
to  the  point  of  application  of  the  load  P  ;  in  other  words, 
the  load  is  applied  at  one  extremity  of  a  diameter  and  the 
neutral  axis  is  tangent  to  the  curve  at  the  other  extremity 
of  that  diameter. 

In  Fig.  i  if  the  load  P  is  applied  at  Q'  (on  the  curve) 
the  neutral  axis  N'Bf  will  be  tangent  to  the  ellipse  at  B', 
the  other  extremity  of  the  diameter  Q'B'. 

If  the  point  of  application  of  the  force  P  moves  along 


Art.  115.]  ECCENTRIC  LOADING.  729 

the  indefinite  straight  line  BQ,  the  coordinates  x'  and  yf 

x' 
will  vary  in  the  same  proportion,  making  —  a  constant. 

From  eq.  (2) : 

%*??% (5) 

x' 
Hence,  as  —  is  constant,  all  neutral  axes  will  be  parallel 

while  the  point  Q  moves  along  a  straight  line. 

Again,  the  coordinates  x  and  y  of  the  points  of  inter- 
section of  the  line  QB  with  the  neutral  axes  must  neces- 
sarily be  opposite  in  sign  from  x'  and  /,  as  the  origin  C 
lies  between  them.  If  therefore  —  x  and  —  y  be  inserted 
in  eq.  (2) : 

^+^J  =  i (6) 

By  similarity  of  triangles,  a  being  a  constant: 

-'  =  -=a     /.   xx'=ay'x=ayy'.      ...     (7) 

y    y 

Eq.  (7)  in  connection  with  eq.  (6)  shows  that  each  of 
the  quantities  x'x  and  y'y  is  constant,  and  that  is  equivalent 
to  making  the  products  of  the  segments  of  the  line  QB  on 
either  side  of  C  constant : 

QCxCB=QfCxCB'=r'*=Q"CxCB".     .     .     (8) 

As  the  point  of  application  of  P  will  always  be  given,  the 
quantity  to  be  found  will  be  the  distance  from  the  centre  C 
to  the  neutral  axis,  which  may  be  called  v.  The  semi- 
diameter  r'  =  CQ'  at  once  becomes  known  after  the  ellipse 
of  inertia  is  constructed.  In  general,  therefore: 


(9) 


730  MISCELLANEOUS  SUBJECTS.  [Ch.  XVI. 

In  some  cases  the  reverse  problem  is  given,  i.e.,  v  is 
known  and  the  distance  of  the  point  of  application  of  the 
load  P  is  required.  Hence, 

r'2 

QC=— (10) 

Rotation  of  the  Neutral  Axis  about  a  Fixed  Point  in  It. 

One  feature  of  eq.  (2)  remains  to  be  considered  before 
the  actual  application  of  the  preceding  results  can  be  made 
to  form  a  complete  graphical  construction.  If  the  co- 
ordinates x  and  y  of  the  neutral  axis  be  considered  constant, 
while  the  coordinates  xr  and  y'  of  the  point  of  application 
of  the  load  P  vary,  eq.  (2)  shows  that  the  path  of  the  move- 
ment of  the  point  of  application  of  P  will  be  a  straight 
line,  since  the  equation  is  of  the  first  degree  in  respect 
to  x'  and  y' .  This  is  equivalent  to  a  movement  of  rota- 
tion of  the  neutral  axis  about  the  fixed  point  whose  coor- 
dinates are  x  and  y,  while  x'  and  y'  determine  the  path 
through  which  the  line  of  action  of  P  moves.  The  same 
result  can  be  shown  by  treating  eq.  (i)  in  precisely  the  same 
manner  for  a  fixed  or  constant  value  of  k,  that  constant 
being  zero  for  the  neutral  axis. 

The  preceding  procedures  may  be  applied  to  a  number 
of  problems,  one  or  two  of  which  will  be  illustrated.  It  is 
sometimes  desired  to  determine  that  part  of  the  cross-sec- 
tion of  a  member  of  a  structure,  or  sometimes  of  the  struc- 
ture itself,  within  which  a  resultant  load  may  be  applied 
anywhere  without  any  change  in  the  kind  of  stress  induced, 
usually  compression. 

Application  of  Preceding  Procedures  to  Z-bar  and  Rectangular 

Sections. 

Let  it  be  required  to  ascertain  within  what  part  of  a 
Z-bar  section  an  axial  compressive  force  may  be  applied 
without  any  part  of  the  section  being  subjected  to  tensile 


Art.  115.]  ECCENTRIC  LOADING.  731 

stress.  The  Z-bar  section  is  shown  in  Fig.  2,  the  depth 
of  bar  being  6  inches  and  the  thickness  of  metal  f  inch. 
As  this  section  is  unsymmetrical  the  axes  for  the  principal 
moments  of  inertia  passing  through  the  centre  of  gravity 
C  of  the  section  will  be  inclined  to  the  central  plane  of 
the  web.  The  ellipse  of  inertia  MVNU  has  MN  for  its 
major  axis  and  UV  for  its  minor  axis,  the  former  representing 
a  moment  of  inertia  of  5  2  and  the  latter  a  minimum  moment 
of  inertia  of  5.7,  the  corresponding  radii  of  gyration  being 
ri=2.55  inches  and  r2=.8i  inch. 

If  no  part  of  the  cross-section  of  the  bar  is  to  be  sub- 
jected to  tension,  the  outer  limits  or  lines  of  that  section 
such  as  TS,  50,  OL,  etc.,  may  be  neutral  axes  for  different 
positions  of  the  load  B,  but  in  no  case  must  the  neutral 
axis  lie  in  any  part  of  the  metal  section,  even  to  cut  across 
a  corner  of  it.  This  means  that  75,  50,  OL,  LH,  HE, 
and  ET  will  be  successively  considered  neutral  axes.  Let 
ET  be  the  first  neutral  axis  considered  or,  rather,  ET  and 
OL  may  be  considered  concurrently,  as  they  are  parallel 
to  each  other  and  at  the  same  distance  from  the  centre  of 
the  ellipse.  First  draw  tangents  to  the  ellipse  parallel  to 
ET  and  OL  as  shown  in  the  figure.  The  points  of  tangency 
will  fix  the  diameter  DA,  which  is  then  extended  to  R 
and  W  in  the  assumed  neutral  axes.  As  shown  in  the 
preceding  demonstration,  the  square  of  half  the  diameter 
represented  by  AD  will  be  equal  to  CR  multiplied  by  CA, 
the  distance  from  the  centre  of  the  ellipse  to  the  point 
of  application  A  of  the  force  P.  The  distance  CR  is  the 
v  of  eq.  (10),  while  CA  is  the  distance  QC  desired,  r'  is 
half  the  diameter  determined  by  the  two  points  of  tangency. 
Dividing  the  square  of  half  the  diameter  by  CR  locates 
the  point  A,  one  of  the  points  desired.  In  precisely  the 
same  manner  D  is  located  by  dividing  the  square  of  half 
the  same  diameter  by  CW  =  CR. 


732  MISCELLANEOUS  SUBJECTS.  [Ch.  XVI. 

Tangents  to  the  ellipse  parallel  to  75  and  HL  are  then 
drawn  as  shown,  one  at  N,  as  indicated,  at  the  lower 
extremity  of  the  ellipse  and  the  other  at  the  upper  extremity, 
thus  locating  the  diameter  NCF.  Squaring  half  the  diam- 
eter so  determined  and  then  dividing  by  the  distance  from 
the  centre  of  the  ellipse  to  TS  or  HL  along  the  diameter 
NF,  the  points  B  and  F  are  found.  In  a  precisely  similar 
way  the  vertical  tangents  indicated  are  drawn  parallel  to 
SO  and  EH,  determining  the  corresponding  diameter.  By 
the  use  of  that  diameter  in  the  manner  already  indicated, 
the  points  G  and  K  are  located. 

The  points  A  and  B  are  points  of  application  of  the  force 
P  when  ET  and  TS  respectively  are  neutral  axes.  In  the 
preceding  sections  of  this  article  it  has  been  shown  that  if 
a  neutral  axis  such  as  ET  be  revolved  about  a  point  in  it, 
as  T,  to  the  position  TS,  the  corresponding  path  of  the  point 
of  application  of  the  load  will  be  a  straight  line,  and  in  this 
case  AB  will  be  that  straight  line,  since  the  two  points  A 
and  B  correspond  to  the  neutral  axes  ET  and  TS.  By 
similarly  connecting  the  other  points,  the  closed  figure 
ABKDFG  is  found.  So  long  as  the  force  P  acts  within 
this  area  no  part  of  the  section  can  be  subjected  to  tension, 
but  if  the  point  of  application  is  outside  of  this  figure 
some  part  of  the  section  will  be  in  tension 

The  closed  figure  thus  established  is  called  the  "  core 
section."  Although  it  possesses  much  analytic  interest, 
the  ordinary  operations  of  the  engineer  are  such  as  to  make 
it  of  comparatively  little  value  in  actual  structural  opera- 
tions. 

If  any  line  such  as  Z'L  parallel  to  a  tangent  to  the 
ellipse  at  g  be  drawn  through  a  corner  L  of  the  Z-bar  sec- 
tion, and  if  a  line  dgZ'  be  drawn  through  the  same  point 
of  tangency  and  the  centre  C,  cutting  the  side  of  the  core 
at  d,  it  is  shown  in  the  preceding  section  of  this  article 


Art.  115-] 


ECCENTRIC  LOADING. 


733 


that  the  product  of  dC  by  CZ'  is  equal  to  the  square  of  the 
semi-diameter  Cg  of  the  ellipse  of  inertia  For  any  other 
position  of  a  line  Z'L  the  same  general  observation  holds, 
the  line  always  being  parallel  to  a  tangent  to  the  ellipse 


FIG.  2  * 


at    a   point  through  which  is  drawn  the  line  extending 
through  the  centre  and  cutting  the  side  of  the  core. 

Probably   the  most   usual   section   to   which   the   core 

*  A  number  of  construction  lines  shown  in  this  figure  are  drawn  for  use 
in  the  next  article. 


734 


MISCELLANEOUS  SUBJECTS. 


[Ch.  XVI. 


procedure  may  be  applied  is  the  simple  rectangle.  A 
masonry  structure  having  such  a  horizontal  section  must 
be  designed  so  that  compression  only  may  always  be 
found  in  it.  A  simple  diagram  of  pressures  will  show  that 
the  resultant  force  or  load  must  act  within  the  middle 
third  of  the  section,  but  Fig.  3  shows  the  core  procedure 
applied  to  the  same  axis.  A B  is  the  length  of  the  section 
and  BD  is  the  width.  AB  is  usually  taken  as  one  unit. 
The  ellipse  OLMN  is  drawn  with  its 
semi-major  axis  LC  representing  the 
greatest  radius  of  gyration  of  the  rec- 
tangle and  the  semi-minor  axis  OC  is 
laid  off  equal  to  the  least  radius  of 
gyration.  Two  lines  drawn  tangent 
to  the  ellipse  at  M  and  N  parallel  to 
BD  and  ED  will  determine  the  axes 
of  the  ellipse,  in  fact  already  known, 
then  dividing  the  square  of  each  semi- 
axis  by  the  normal  distance  of  C  from 
BD  and  ED,  respectively,  the  dis- 
tances CF  and  CK  will  be  found,  thus 
fixing  the  points  F  and  K.  The  points 
H  and  G  are  found  in  precisely  the  same  manner,  using  the 
sides  AE  and  AB  respectively.  As  already  indicated,  the 
distance  of  H  from  BD  will  be  one-third  of  AB,  while  K 
will  be  one-third  of  BD  from  AB. 

General  Observations. 

The  preceding  results  show  that  bending  combined  with 
uniform  stress  induced  by  a  load  normal  to  the  section 
will  prevent  the  neutral  axis  from  passing  through  the  centre 
of  gravity  of  the  cross-section.  Furthermore,  in  this  general 
case  the  neutral  axis  or  neutral  surface  will  not  be  at  right 
angles  to  the  plane  containing  the  axis  of  the  piece  and  the 


FIG.  3. 


Art.  116.]  GENERAL  FLEXURE  TREATED  BY  CORE  METHOD.        735 

line  of  action  of  the  force  unless  that  plane  contains  one 
of  the  principal  axes  of  inertia. 

Manifestly  the  neutral  axis  for  any  section  will  be  on 
the  opposite  side  of  the  centre  of  gravity  of  that  section 
from  the  force  P.  Eq.  (8)  shows  that  if  the  force  acts  at 
C,  making  QC  equal  zero,  CB  will  be  infinitely  great,  which 
means  that  the  stress  will  be  uniformly  distributed,  i.e., 
there  will  be  no  bending.  On  the  other  hand,  if  the  force 
P  is  at  an  indefinitely  great  distance  from  C,  making  QC 
infinity,  then  will  CB  be  equal  to  zero,  i.e.,  the  neutral 
axis  will  pass  through  the  centre  of  gravity.  This  is  the 
ordinary  case  of  flexure  and  it  is  equivalent  to  taking  all 
load  on  the  member  at  right  angles  to  its  axis. 

Art.  116. — General  Flexure  Treated  by  the  Core  Method. 

The  procedures  given  in  the  preceding  article  may  be 
used  for  the  general  problem  of  flexure  for  straight  beams 
of  any  form  of  cross-section  carrying  any  parallel  loads  at 
right  angles  to  their  axes,  the  loads  supposed  to  be  acting 
in  a  plane  which  contains  the  axis  of  the  beam  in  each  case. 
Under  such  conditions  there  will  clearly  be  no  direct  uni- 
form compression  on  any  normal  section  of  a  beam.  This 
is  equivalent  to  assuming  that  the  flexure  is  produced  by  an 
indefinitely  small  force  acting  parallel  to  the  axis  (or  at 
right  angles  to  a  normal  section)  of  the  beam  and  at  an 
infinite  distance  from  the  latter. 

It  is  clear,  since  the  product  of  the  distance  of  the  point 
of  application  of  a  force  normal  to  the  cross-section  from 
the  centre  of  gravity  of  the  latter  multiplied  by  the  disr 
tance  of  the  neutral  axis  from  the  same  point,  but  on  the 
opposite  side  from  the  point  of  application  of  the  loading, 
must  be  equal  to  the  square  of  half  the  diameter  of  the 
ellipse  of  inertia,  that  if  that  square  be  divided  by 


736  MISCELLANEOUS  SUBJECTS.  [Ch.  XVI. 

infinity,  the  distance  of  the  point  of  application  of  the 
load  from  the  centre,  the  quotient  will  be  zero,  i.e.,  the 
neutral  axis  must  pass  through  the  centre  of  gravity  of 
the  section. 

This  condition  is  further  equivalent  to  taking  any  finite 
loading  at  right  angles  to  the  axis  of  the  beam,  as  in  the 
ordinary  cases  of  engineering  practice.  The  stresses  found 
in  the  normal  section  in  such  cases  will  be  the  direct  tension 
and  compression  with  intensity  varying  directly  as  the 
normal  distance  from  the  neutral  axis  with  the  accompany- 
ing shears,  as  in  the  common  theory  of  flexure. 

The  preceding  investigations  show,  however,  that  with 
unsymmetrical  sections  the  neutral  axis,  while  passing 
through  the  centre  of  gravity  of  the  section,  is  not  at  right 
angles  to  the  plane  of  loading,  unless  that  plane  happens 
to  contain  one  of  the  two  principal  axes  of  inertia  of  the 
section. 

Let  the  Z-bar  section  shown  in  Fig.  2  of  the  preceding 
article  be  considered  and  suppose  that  the  loading  acts  in 
the  vertical  plane  ZZ' ',  the  latter  line  passing  through  the 
centre  of  gravity  C  of  the  cross-section.  It  may  be  con- 
sidered that  the  Z-bar  is  supported  at  each  and  on  the  lower 
surface  HL  of  the  lower  flange.  Inasmuch  as  the  bending 
moment  acts  in  the  plane  ZCZ'  the  neutral  axis  will  be 
drawn  through  the  centre  C  parallel  to  the  tangents  to  the 
ellipse  where  the  line  ZZ'  cuts  the  latter,  as  shown  at  g 
and  at  the  opposite  end,  not  lettered,  of  the  vertical  diameter. 
The  diameter  A'B'  is  then  the  neutral  axis  desired.  The 
line  Cb  drawn  at  right  angles  to  ZZ'  may  be  considered 
the  axis  of  the  external  bending  moment  to  which  the  beam 
is  subjected.  The  angle  between  Cb  and  the  neutral 
axis,  is  a,  as  shown. 

If  the  coordinate  x  be  taken  as  at  right  angles  to  the 
neutral  axis  A'B't  and  if  dA  represent  an  element  of  the 


Art.   116.]  GENERAL  FLEXURE    TREATED   BY  CORE   METHOD.        737 

normal  section  of  the  beam,  then  the  distance  of  that  element 
from  the  neutral  axis  measured  parallel  to  ZZ'  will  be 
x  sec  a.  If  k  is  the  maximum  intensity  of  stress  at  any 
point  of  the  section,  that  stress  will  occur  at  L  or  T,  where 
the  value  of  x=n  is  the  greatest  for  the  entire  section. 
The  distance  of  that  point  parallel  to  ZZ'  will  be  n  sec  a. 
If  M  is  the  value  of  the  external  bending  moment  acting 
in  the  plane  ZZ',  dM  may  be  written: 

k 

dM  =  - x  sec  a  -  dA  -  x  sec  a.  (i) 

wseca 

If  /  is  the  moment  of  inertia  of  the  section  about  the 
neutral  axis, 


M 


=  CdM  =-I  sec  a-  =-r'A  sec  a.  ,  (2) 

J  n  n 


In  Fig.  2  the  line  ZZ'  cuts  at  d  the  side  DF  of  the  core. 
Let  the  distance  dC  be  represented  by  /.  Then,  as  shown 
in  the  preceding  article, 

jCZ'=Cg. 

But  the  radius  of  gyration  of  the  section  about  the  axis 
A'B'  has  been  shown  in  Art.  81  to  be  equal  to  the  normal 
distance  r'  between  the  neutral  axis  and  the  parallel  tangent 
to  the  ellipse  drawn  at  g. 

Cg  =  r'  sec  a. 

It  has  already  been  seen  that  CZf  is  equal  to  n  sec  a. 
r'2  sec  a.  =  nj. 

If  this  value  of  r'2  sec  a  be  substituted  in  the  third 
member  of  Eq.  (2)  there  will  result, 

M=kAj.  (3) 


738  MISCELLANEOUS  SUBJECTS.  [Ch.  XVI. 

Eq.  (3)  is  the  expression  for  the  external  bending  moment 
in  terms  of  the  greatest  intensity  of  stress  in  the  section, 
the  area  of  that  section,  and  the  distance  /from  the  centre 
of  the  section  to  the  side  of  the  core  as  constructed  by  the 
methods  explained  in  the  preceding  section.  Although  the 
construction  has  been  made  with  the  Z  section  the  method 
of  procedure  is  precisely  the  same  for  any  form  of  section 
whatever. 

Component  Moments. 

By-  again  referring  to  Fig.  (2)  of  the  preceding  article 
it  will  be  seen  that  M  cos  a  is  that  component  of  the  external 
moment  whose  axis  is  parallel  to  the  neutral  axis,  while 
the  component  M  sin  a  has  an  axis  be  at  right  angles  to  the 
neutral  axis,  but  lying  in  the  plane  of  the  normal  section 
of  the  beam.  The  former  component  produces  the  bending 
stresses  about  the  neutral  axis,  the  maximum  intensity  of 
which  is  k  and  a  deflection  normal  to  it;  the  latter  compo- 
nent moment  tends  to  produce  an  oblique  movement  of 
the  beam  in  consequence  of  its  unsymmetrical  section. 

This  tendency  in  oblique  flexure,  especially  marked  with 
unsymmetrical  sections,  is  always  toward  that  position 
in  which  the  least  radius  of  gyration  of  the  section  (repre- 
sented by  the  least  semi-axis  of  the  ellipse  of  inertia)  is 
found  in  the  plane  of  bending,  i.e.,  that  plane  in  which  the 
bending  moment  acts. 

In  Fig.  2  of  the  preceding  article  ZZf  is  the  plane  in  which 
the  vertical  loading  acts,  and  it  is  clear  that  the  plane  in 
which  the  resultant  bending  compression  on  one  side  of  the 
neutral  axis  A'B'  and  the  resultant  bending  tension  on  the 
other  side  act  is  not  the  plane  in  which  ZZ'  lies,  but  inclined 
somewhat  to  the  right  of  CZ.  Inasmuch  as  these  two 
planes  are  neither  the  same  nor  parallel,  there  must  be 
combined  with  the  couple  producing  pure  flexure  such  a 


Art.  117.]  PLANES  OF   RESISTANCE  IN  OBLIQUE   FLEXURE.         739 

couple  as  to  make  the  resultant  external  moment  equal 
and  opposite  to  the  internal  resisting  moment,  and  the 
component  of  M  represented  by  be  is  such  a  couple,  Ce 
representing  the  couple  producing  pure  flexure  about 
A'B'. 

These  analytic  considerations  show  how  essential  it  is 
to  give  careful  consideration  to  the  principles  governing 
oblique  or  general  flexure  for  loads  not  in  a  plane  of 
symmetry  of  a  beam  and  for  unsymmetrical  sections. 

The  method  of  finding  the  location  of  the  plane  of 
resistance  of  the  bending  stress  existing  in  any  normal 
section  of  the  beam  will  be  given  in  the  next  article. 

Art.  117. — Planes  of  Resistance  in  Oblique  or  General  Flexure. 

The  preceding  treatment  of  general  flexure  has  shown 
that  the  plane  of  action  of  the  external  bending  moment 
will  not  in  general  coincide  with  the  plane  in  which  the 
internal  resisting  couple  acts.  The  plane  of  the  external 
bending  moment  is  supposed  to  pass  through  the  axis  of 
the  beam  assumed  to  be  straight.  If  this  external  bend- 
ing couple  is  to  produce  pure  flexure  it  must  .be  in  equilib- 
rium with  the  internal  moment  produced  by  the  stresses 
in  any  normal  section,  and  that  requires  that  the  two  planes 
of  action  shall  either  coincide  or  be  parallel. 

Let  it  be  supposed  that  the  6X3^Xf-inch  steel  angle 
section  shown  in  Fig.  i  represent  any  unsymmetrical  sec- 
tion, and  let  it  also  be  supposed  that  GY  is  the  neutral 
axis  of  the  section,  G  being  the  centre  of  gravity;  then  let 
GX  and  GY  be  the  axes  of  rectangular  coordinates  negative 
when  measured  to  the  left  and  downwards.  The  stresses 
above  GY  will  be  supposed  compressive,  and  those  below, 
tensile.  The  intensities  will  be  assumed  to  vary  directly 
as  the  normal  distances  from  GY  as  in  the  ordinary  theory 


740  MISCELLANEOUS  SUBJECTS.  [Ch.  XVI. 

of  flexure.  The  centre  of  all  the  compressive  stresses  will 
be  taken  at  C  and  at  T  for  the  tensile  stresses.  The  plane 
whose  trace  is  CT  will  be  called  the  plane  of  resistance, 
while  AB'  will  be  taken  as  the  plane  of  action  of  the 
external  bending  moment.  In  other  words,  if  the  angle 
were  to  carry  vertical  loading  as  a  beam  AB  should  be 
vertical  with  the  lines  of  cross-section  correspondingly 
inclined. 

If  x±  and  TI  are  the  coordinates  of  the  centre  C  of  the 
compressive  stresses  in  the  section  and  if  a  is  the  intensity 
of  stress  at  a  unit's  distance  from  the  neutral  axis  GY, 
eqs.  (i)  and  (2)  will  immediately  result: 

ffyaxdxdyjfxydxdy^ 
ffaxdxdy      ffxdxdy      Qi 

ffxaxdxdy    ffx2dxdy     j't 
~~  ffaxdxdy       ffxdxdy      Qi 

The  quantities  J\  and  J'i  are  the  so-called  "product 
of  inertia  "  and  the  moment  of  inertia  of  that  part  of  the 
cross-section  lying  above  GY,  while  Q\  is  obviously  the 
statical  moment  of  the  same  part  of  the  cross-section  in 
reference  to  the  same  axis. 

If  the  subscript  2  be  used  for  the  corresponding  quanti- 
ties relating  to  that  part  of  the  section  below  GY,  eqs.  (3) 
and  (4)  will  at  once  result,  the  negative  sign  being  used 
in  the  second  member  because  the  coordinates  are  negative: 


Art.  117.]    PLANES   OF  RESISTANCE  IN   OBLIQUE    FLEXURE.      741 

Q,  I  and  /  represent  quantities  belonging  to  the  whole 
cross-section,  then,  since  G  is  the  centre  of  gravity  of  that 
section, 


e 


\ 

\ 

\ 

••»    \ 

-.555\ 

\ 


FIG.  i. 


It  is  desired  to  find  the  straight  line  joining  C  and  T, 
and  in  order  to  do  that  the  general  equation  of  a  straight 
line  may  be  written  as  follows: 


x+by-=c. 


(5) 


742  MISCELLANEOUS  SUBJECTS.  [Ch.  XVI. 

If  y\  and  xi  taken  from  eqs.  (i)  and  (2)  be  first  written 
in  eq.  '(5)  and  then  y2  and  x2  from  eqs.  (3)  and  (4),  and  if 
the  second  of  the  equations  so  formed  be  subtracted  from 

the  first,  there  will  result:     b  =  — — . 
Then  eq.  (5)  will  take  the  form 

x=jy+c.  .../•£.        ....     (6) 

In 'Fig.  i  suppose  a  line  parallel  to  CT  drawn  from  G 
to  B.  If  the  ordinate  x\  be  produced  upward,  the  line 
BC  =  Gc'  will  be  determined.  If  in  eq.  (6)  y  =  o,  x  —  c  =  Gc'  = 
BC.  The  triangles  with  the  bases  y\  and  y2  will  then  be 
similar  and  that  similarity  will  be  expressed  by  the  following 
equation,  remembering  that  x  and  y  are  negative: 

^-=*¥-    -     -  (7) 


Substituting  the  values  of  xi,  yi,  X2  and  y2  established 
above  there  will  result  the  following  value  of  c: 


_ 


JQ 

Placing  this  value  of  c  in  eq.  (6), 


This  is  the  equation  of  the  line  CT,  Fig.  i  ,  drawn  through 
the  centres  of  the  tensile  and  compressive  resisting  stresses 
acting  in  the  normal  section,  i.e.,  it  is  the  trace  of  the  plane 
in  which  the  resisting  couple  acts.  The  tangent  of  the 


Art.  117.]    PLANES  OF   RESISTANCE   IN  OBLIQUE  FLEXURE.        743 


angle  which  it  makes  with  the  neutral  axis  GY  is  -^=-. 

dy    J 
If  GY  is  one  of  the  principal  axes  of  inertia  of  the  section 

dx 

J  =o  and  —  becomes  infinitely  great,  i.e.,  in  that  case 

the  line  CT  is  at  right  angles  to  GY  and  it  will  presently 
be  shown  that  it  will  pass  through  G,  the  centre  of  gravity 
of  the  section. 

If  y=o  in  eq.  (8), 


The  distance  Gc'  is  on  the  negative  side  of  G.     Again  if 
#=o,  there  will  result: 


These  coordinates  Ge  and  Gc'  shown  in  Fig.  i  give  two 
points  e  and  c'  in  the  desired  line  CT,  which  must  agree 
obviously  with  the  points  C  and  T  as  found  by  computa- 
tions. 

If  Ge  should  be  zero,  eq.  (n)  will  result: 


(n) 


Inasmuch  as  the  moments  of  inertia  I'\  and  I'  2  will 
always  have  real  values  for  an  actual  section,  in  general 
if  eq.  (n)  holds  true,  then  must  /i=/2=o.  That  condi- 
tion will  of  course  exist  for  the  principal  axes  and  for  the 
case  where  at  least  one*  of  the  coordinate  axes  is  an  axis 
of  symmetry  of  the  section. 

Although  the  figure  used  for  the  establishment  of  the 
preceding  formulae  is  the  normal  section  of  a  steel  angle, 
those  formulae  are  completely  general  and  are  applicable 


744  MISCELLANEOUS  SUBJECTS.  [Ch.  XVI. 

to  any  form  of  cross-section  whatever,  as  indicated  by 
eqs.  (i)  and  (2)  and  all  the  equations  following. 

It  is  thus  seen  that  if  the  plane  of  action  A B  of  any 
external  loading  producing  flexure  of  a  beam  with  unsym- 
metrical  cross-section  is  parallel  to  the  plane  whose  trace 
is  CT,  there  will  be  pure  bending  only  as  the  external  bend- 
ing moment  has  the  same  axis  as  the  couple  formed  by  the 
internal  stresses.  The  planes  of  the  external  bending 
moment  and  that  of  the  internal  resisting  stresses  may  in 
some  cases  coincide. 

If  the  steel  angle  shown  in  Fig.  i  is  to  act  as  a  beam 
under  vertical  loading  in  pure  flexure,  the  end  supports 
should  be  so  formed  as  to  make  the  lines  AB  and  CT  verti- 
cal. In  general,  whatever  may  be  the  cross-section  of  a 
beam,  the  latter  should  be  so  held  at  its  points  of  support 
that  the  loading  will  produce  pure  flexure.  If  the  section 
of  the  beam  has  an  axis  of  symmetry,  the  plane  of  loading 
may  be  taken  through  the  axes  of  symmetry  of  the  cross- 
section. 

Example.  The  application  of  the  preceding  formulae 
may  be  illustrated  by  using  the  6X3|-inch,  22.4-lb.  steel 
angle  shown  in  Fig.  i.  The  thickness  of  each  leg  is  .75 
inch.  By  using  eqs.  (i)  and  (2)  there  will  at  once  result: 

7'i=g.4i;        /'2  =  i3-94;        /i=5-47; 
72=3-04;          /  =  8.5i;  0  =  5.484 

Inserting  these  values  in  eqs.  (i),  (2),  (3)  and  (4)  there 
will  result : 

3/1=1  in.;  #1=1.72  ins.;  y2  =  -.555  in.; 

#2=  —2. 54 ins.;     XQ  =  —1.02  ins. ;         y0  =  .3 72  in. 

These  coordinates  are  laid  off  in  Fig.  i,  as  shown,  so  as 
to  locate  the  four  points  C,  e,  c'  and  T.  In  making  these 


Art.  118.]  DEFLECTION  IN  OBLIQUE  FLEXURE.  745 

computations  it  should  be  remembered  that  I'\  and  I' 2 
are  moments  of  inertia  of  areas,  one  of  whose  sides  coin- 
cides with  the  axis  of  y  and  that  the  same  observation  is 
also  true  of  the  quantities,  J\  and  /2,  as  well  as  Q. 


Art.  118. — Deflection  in  Oblique  Flexure. 

The  general  case  of  deflection  of  a  beam  with  unsym- 
metrical  cross-section,  or  of  a  beam  with  symmetrical 
cross-section  but  loaded  obliquely,  may  readily  be  found 
by  the  aid  of  the  ordinary  formulae  for  flexure  used  in 
connection  with  the  preceding  investigations.  The  requisite 
treatment  may  be  well  illustrated  by  considering  the  case 
of  a  6  Xsi  Xf-inch  steel  angle,  the  section  of  which  is  shown 
in  Fig.  i  to  be  same  as  that  used  in  the  preceding  article. 
Such  an  angle  may  be  considered  to  be  used  as  a  beam 
in  roof  work  or  for  some  other  similar  purpose  with  the 
6-inch  leg  placed  in  a  vertical  position.  It  will  be  assumed 
that  the  span  length  is  15  feet  =  i8o  inches  and  that  the 
angle  is  to  carry  as  a  beam  a  uniform  load  of  200  pounds 
per  linear  foot.  The  data  given  in  an  ordinary  handbook 
on  steel  sections  will  show  the  position  of  the  centre  of 
gravity  G  of  the  section  and  enable  the  ellipse  of  inertia  to 
be  constructed  as  in  Fig.  i. 

The  maximum  radius  of  gyration  represented  by  the 
greater  semi-axis  of  the  ellipse  is  1.97  inches,  while  the  least 
radius  of  gyration  at  right  angles  to  the  preceding  and 
represented  by  the  smaller  axis  is  .75  inch.  The  load 
acts  in  a  vertical  plane  passing  through  the  axis  G.  The 
various  dimensions  of  the  cross-section  required  in  the 
computations  are  all  shown  in  Fig.  i. 

By  drawing  vertical  tangents  on  opposite  sides  of  the 
ellipse,  the  neutral  axis  A'B'  drawn  through  the  points  of 
tangency  and  the  centre  G  of  the  ellipse  is  determined. 


746  MISCELLANEOUS  SUBJECTS.  [Ch.  XVI. 

This  neutral  axis  of  the  section  makes  the  angle,  46°  30', 
as  carefully  measured  on  the  diagram,  with  the  horizontal 
axis  of  Y.  By  drawing  a  tangent  to  the  ellipse  parallel 
to  A'B'  the  radius  of  gyration  about  the  neutral  axis  is 
found  to  be  i.i  inches,  i.e.,  the  normal  distance  between 
the  neutral  axis  and  the  parallel  tangent  to  the  ellipse. 

The  greatest  deflection  of  the  angle  beam  will  be  found 
at  the  centre  of  span  at  which  the  moment  of  the  external 
forces  is 

M  =  2°°X225Xi2  =67,500  in.-lbs.  (i) 


The  component  moment,  as  shown  in  the  preceding 
article,  with  axis  parallel  to  the  neutral  surface,  is 

M  cos  a  =  .6884M  =  46, 467  in.-lbs.      ...     (2) 

The  component  moment  having  an  axis  at  right  angles 
to  the  neutral  axis  is,  similarly, 

M  sin  a  =  .7 2 5471^=48,964  in.-lbs.      ...     (3) 

The  actual  flexure  is  produced  by  the  first  of  these  com- 
ponents M  cos  a.  The  deflection  produced  by  it  will  obvi- 
ously-be  normal  to  the  neutral  axis,  and  it  can  be  computed 
by  the  ordinary  formula  for  the  deflection  at  the  centre 
of  span  of  a  beam  simply  supported  at  each  end  and  loaded 
uniformly  throughout  its  length,  the  uniform  load  to  be 
taken  in  this  case  as  200  cos  01  =  138  pounds  per  linear  foot. 
If  g  is  the  load  per  linear  foot  of  span,  the  usual  expres- 

1-^4 

sion  for  the  centre  deflection  is  w  =        ~r-     Substituting 

3&4&Z 

138X15  for  gl  in  the  formula,  /  =  180  inches,  E  =30,000,000, 


Art.  1 1 8.1 


DEFLECTION  IN   OBLIQUE  FLEXURE. 


747 


and  I  =  7.94  (moment  of  inertia  of  section  about  the  neutral 
axis)  there  will  result: 


w=o.66  inch. 


(4) 


As  cos  a  =  .6 884  and  sin  a  =  .7 2  54,  the  vertical  deflec- 
tion =.66  X.  6884  =.454  inch;  and  the  horizontal  deflec- 
tion =  .66  X. 72 54  =.479  inch. 


FIG.  i. 

It  is  thus  seen  that  the  horizontal  deflection  slightly 
exceeds  the  vertical,  in  consequence  of  the  major  axis  of 
the  ellipse  of  inertia  being  slightly  inclined  to  a  vertical 


748  MISCELLANEOUS  SUBJECTS.  [Ch.  XVI. 

line,  thus  causing  the  inclination  of  the  neutral  axis  of 
the  section  to  be  relatively  large. 

Precisely  the  same  general  treatment  would  be  followed 
for  any  form  of  cross-section  or  any  other  amount  or  dis- 
position of  loading. 

In  the  preceding  article  where  the  same  angle  was  so 
held  as  to  make  the  plane  of  loading  parallel  to  that  of 
the  resisting  couple,  the  horizontal  diameter  of  the  ellipse 
drawn  through  G  is  the  neutral  axis  corresponding  to  the 
conjugate  diameter  DF,  parallel  to  the  trace  of  the  plane 
of  the  resisting  internal  couple  as  determined  in  that 
article.  The  normal  distance,  1.95  inches,  between  the  hori- 
zontal diameter  through  G  and  the  horizontal  tangent  at 
F  is  the  radius  of  gyration  corresponding  to  the  horizontal 
neutral  axis  through  G.  As  the  area  of  cross-section  of 
the  steel  angle  is  6.56  square  inches,  the  moment  of  inertia 
corresponding  to  the  horizontal  neutral  axis  through 
G  is  I  =  6.56  X  i.  952  =24.93,  the  moment  of  inertia  of  the 
cross-section  about  the  neutral  axis  A'B't  Fig.  i,  is 
7  =  6.56  Xi.  i2  =  7.94.  The  distance  from  the  horizontal 
neutral  axis  through  G  to  the  extreme  fibre  is  3.82  inches, 
while  the  corresponding  distance  of  the  extreme  fibre  from 
A'Bf  is  2.3  inches.  Hence,  the  resisting  moment  for  the 
horizontal  neutral  axis  through  G  is 


3.82 

For  the  neutral  axis  A'B': 


2-3 
Hence  — -f  =  1.9.     In  other  words,  the  same  angle  placed 


Art.  119.]       ELASTIC  ACTION  UNDER  DIRECT  LOADING.  749 

so  as  to  take  the  vertical  loading  in  a  plane  parallel  to  the 
resisting  internal  couple  will  offer  nearly  twice  as  much 
bending  "resistance  with  the  same  extreme  fibre  stress  as 
when  placed  with  the  longer  leg  vertical.  Economic  use 
of  the  metal  as  well  as  avoidance  of  unnecessary  deflection, 
therefore,  requires  that  the  beam  of  unsymmetrical  section 
shall  be  so  held  at  its  supports  as  to  make  the  plane  of 
loading  parallel  to  the  resisting  plane  and  as  nearly  parallel 
to  the  greater  axis  of  the  ellipse  of  inertia  of  the  cross- 
section  as  possible. 


Art.  119. — Elastic  Action  under  Direct  Loading  of  a  Composite 
Piece  of  Material. 

Let  it  be  supposed  that  a  combined  straight  or  cylin- 
drical piece  of  material  with  length  L  is  subjected  to  the 
direct  stress  of  either  tension  or  compression.  If  the  total 
area  of  cross-section  is  A,  it  may  be  assumed  to  be  composed 
of  the  following  parts : 

A i  =area  of  cross-section  with  modulus  of  elasticity  E\\ 
A2=  area  of  cross-section  with  modulus  of  elasticity  £2 ; 

A  3  =  area  of  cross-section  with  modulus  of  elasticity  £3 ; 
etc.,  etc. 

Then  will 

A=Ai+A2+A3+etc.    .     .     .>*,.-     (i) 

Let  the  total  load  P  act  parallel  to  L  and  let  /  be  the 
strain  per  unit  of  length  of  the  piece,  i.e.,  the  unit  strain, 
then  will  IL  be  the  total  lengthening  or  shortening  of  the 
piece.  Under  these  conditions  every  part  of  the  piece 
will  be  subjected  to  the  same  rate  of  longitudinal  strain 
and  the  following  equation  may  be  at  once  written: 


750                               MISCELLANEOUS  SUBJECTS.  [Ch.  XVI, 

EilA1+E2lA2+E3lA3+etc.=P=ElA.  .     .     (2) 
Hence, 

,           _  P  _  ,      v 


Also  the  first  and  third  members  of  eq.  (2)    will  give 
eq.  (4): 

-    ElAi+E2A2+E3A3+etc.  , 

-  -       '     ' 


Eq.  (3)  will  give  the  lengthening  or  shortening  of  each 
unit  of  length  of  the  piece  under  any  assigned  load  P,  the 
moduli  of  elasticity  of  the  areas  of  the  different  parts  of 
the  section  being  known. 

The  modulus  of  elasticity  E  given  by  eq.  (4)  may  be 
considered  a  mean  or  average  modulus  or  an  equivalent 
value  for  the  actual  moduli,  as  the  same  longitudinal  strain 
would  be  yielded  by  a  piece  of  uniform  material  having 
that  modulus  of  elasticity  and  the  same  area  of  cross- 
section  as  the  composite  piece. 

Art.  120.  —  Helical  Spiral  Springs. 

A  spiral  spring  like  that  shown  in  Fig.  i  takes  its  load 
at  the  ends  as  indicated  at  A  and  B.  In  the  general  case 
there  may  be  applied  at  each  end  a  single  load  P  and  a 
couple,  or  either  a  force  or  a  couple  alone  may  act.  The 
analysis  will  be  so  written  as  to  include  concurrent  force 
and  couple  or  either  one  separately.  The  following  nota- 
tion will  be  employed: 

R  =  radius  of  spiral,  Fig.  i  ; 

<j>  =  pitch  angle  of  spiral,  Fig.  i  ; 

z  =  axial  elongation  or  compression  of  spring  under  load- 
ing; 


Art.  120.]  HELICAL  SPIRAL  SPRINGS.  751 

/=  length  of  spiral; 

r  =  radius  of  spiral  wire ; 
P  =  axial  load,  Fig.  i ; 

M=  moment  of  applied  twisting  couple  or  torque,  as- 
sumed to  be  a  right-hand  moment ; 

u  =  unit  strain  at  unit  distance  from  the  neutral  axis 
in  bending  or  flexure ; 

a  =  angle  of  torsion  (unit  strain  at  unit  distance  from 
axis  of  piece  in  torsion) ; 

T=  total  twist  or  rotation  of  spring  measured  on  central 
cylinder  of  spiral ; 

T 

r  =—=  angle  of  twist  of  spring  in  radians. 
J\ 

The  force  P  will  be  considered  positive  when  it  stretches 
the  spring  as  shown  in  Fig.  i.  If  the  force  P  compresses 
the  spring  it  must  have  the  negative  sign  in  all  the  following 
analysis. 

The  moment  M  will  be  considered  a  right-hand  moment 
when  it  twists  the  spiral  so  as  to  bring  the  helical  parts 
near  together,  i.e.,  tightens  the  spiral.  It  should  be  re- 
membered that  all  parts  of  the  spiral  are  uniformly  stressed 
or  bent.  The  cross-section  of  the  spiral  rod  will  be  con- 
sidered circular,  although  the  general  analysis  is  adapted 
to  any  form  of  cross-section. 

The  load  P  produces  a  moment  Mi  about  the  centre  of 
any  section  of  the  spiral  rod  given  by 

Mi  =PR •  .     .     .     (i) 

The  axis  of  this  moment  is  a  horizontal  line  through 
the  centre  of  the  section  and  tangent  to  the  central  cylinder 
of  the  spiral  shown  by  a  broken-line  circle  in  the  lower 
part  of  Fig.  i.  If  A,  Fig.  2,  be  the  centre  of  the  section 


752 


MISCELLANEOUS  SUBJECTS. 


[Ch.  XVI. 


considered,  KL  may  be  taken  as  the  axis  of  the  moment 
PR.  If  AK,  therefore,  represent  by  a  convenient  scale,  the 
moment  Mi  =  PR,  AG  and  GK  (drawn  perpendicular  to 
AG)  will  represent  by  the  same  scale  the  component  mo- 
ments of  Mi  about  those  lines  as  axes  passing  through 
the  centre  of  the  section.  As  the  axis  AG  is  the  axis  of 


FIG.  i 


FIG.  2. 


the  spiral  rod,  it  represents  a  torsion  moment.  Similarly 
GK  represents  a  bending  moment  as  it  lies  in  the  section 
and,  m  fact,  is  a  neutral  axis.  Hence,  if  the  subscripts 
t  and  b  mean  torsion  and  bending, 


And, 


AG=M't=Ml  cos 
GK=M'b=-M  sin 


(2) 
(3) 


Art.  120.]  HELICAL  SPIRAL  SPRINGS.  753 

The  moment  —  M  sin  <£  has  a  negative  sign  because  the 
triangle  AKG,  Fig.  2,  shows  that  it  will  tend  to  untwist 
the  spiral  of  Fig.  i,  which  is  opposite  to  a  positive  effect. 

The  right-hand  moment  M  will  act  at  the  centre  of 
section  of  the  spiral  rod  about  an  axis  parallel  to  AC, 
Fig.  i,  i.e.,  about  BD,  Fig.  2,  and  AB  may  represent  that 
moment.  Its  two  components  will  be: 


M",=Msin  0,        ....  (4) 

AF=M"b=M  cos  0.       ....  (5) 

The  resultant  moments  of  torsion  and  bending  at  the 
section  considered  will  therefore  be: 


Mt=Mi  cos  0+M  sin  <j>,         ...     (6) 
Mb=M  cos  0  —  Mi  sin  0.         ...     (7) 

By  the  common  theory  of  torsion  (correct  for  a  cir- 
cular section  only)  if  G  is  the  modulus  of  shearing  elasticity, 
the  angle  of  torsion,  or  unit  strain  a,  is 


moment     M\  cos  0+M  sin  <£  /ox 

~~  ~~     ~"   '  ' 


TTf4  64 

Evidently,    Q=G  —    (for    circle);     and    Q=G—    (for 
2  6 

square)  . 

If  the  exact  theory  of  torsion  is  used  for  other  sections 
of  the  spiral  rod  than  circular,  the  corresponding  value  of 
a  must  be  introduced,  but  no  other  change  is  needed. 

In  the  same  manner,  if  E  is  the  modulus  of  elasticity 
for  direct  stress,  I  the  moment  of  inertia  of  the  section 


754 


MISCELLANEOUS  SUBJECTS. 


[Ch.  XVI, 


about  its  neutral  axis,  and  if  Q'  =EI  =E —  (for  circular 

4 

section)  or  Q'  =E —  (for  square  section),  the  unit  strain, 
12 

u,  for  bending  is, 

_ moment  _M  cos  0  —Mi  sin  0  ,  , 

The  quantities  a  and  u  are  unit  motions  giving  to  the 
spiral  spring  corresponding  motions  of  rotation  and  axial 
lengthenings  or  shortenings. 

The  torsion  moment  Mt  will  cause  one  end  of  an  indefi- 
nitely short  length  dl  of  the 
spiral  rod  to  rotate  through 
the  angle  adl,  inducing  a 
movement  of  that  end,  rela- 
tive to  the  axis  of  the  spiral, 
perpendicular  to  the  axis  of 
the  rod,  equal  to  Radl,  as 
shown  by  Fig.  3.  The  hori- 
zontal component  of  this 
movement  tangent  to  the 
spiral  cylinder  is,  Radl  sin  0, 
or  for  each  unit  of  length  of 
the  rod,  Ra  sin  0.  As  the 
state  of  stress  is  uniform 
throughout  the  spiral  rod,  the 
twist  of  the  spiral  spring  due  to 


FIG.  3. 


FIG.  4. 


total  circumferential 
torsion  is 


'     P7    c.;^ 
=KLa$m  <j>= 


cos 


sn 


And  the  angle  of  twist  is 


T,_T_lMicos 
R 


sn 


.  f     N 

Sin*,.     (10) 


'    '    '  (loa) 


Art.  120.]  HELICAL  SPIRAL   SPRINGS.  755 

The  axial  component  of  the  same  movement,  as  shown 
by  Fig.  3  is,  Radl  cos  <£.  Hence  the  total  axial  movement 
due  to  torsion  is 

'  _  z?/^1  cos  0  +^f  sm 

z_m  _ 

The  movement  of  a  normal  section  of  the  spiral  rod, 
relative  to  the  axis  of  the  spring,  due  to  bending  about  its 
neutral  axis  parallel  to  GK  and  AF,  Fig.  2,  is  illustrated  by 
Fig.  4.  That  movement  will  be  parallel  to  the  axis  of  the 
rod  and  the  broken-line  triangle  showing  it  and  its  com- 
ponents is  moved  vertically  to  clear  it  from  the  centre  line 
of  the  rod.  The  horizontal  component  representing  the 
tangential  or  rotating  movement  due  to  bending  is  seen  to 
be 

Rudl  cos  0. 

Or,  for  the  entire  length  /  of  the  spring, 

~v,     -pM  cos  0— Mi  sin  <f>  ,     . 

1     =Rl-  -cos<£.     .     .     .     (12) 

The  angular  twist  is 

„     T"     M  cos  4>  —Mi  sin 


Similarly,  the  axial  component  of  the  movement  due 
to  bending,  as  shown  in  Fig.  4,  is 

—  Rudl  sin  </>. 

This  value  is  negative,  as  the  axial  motion  is  downward 
and  opposite  to  that  due  to  torsion  shown  in  Fig.  3. 
Hence, 

„         D7M  cos  <£  —  Mi  sin  </>    . 
z    =  -Rl-        -g-          -an*.     .     . 


756  MISCELLANEOUS  SUBJECTS.  [Ch.  XVI. 

The  angle  of  twist  of  the  spring  under  loading  will  be 
the  sum  of  the  second  members  of  eqs.  (ioa)  and  (i2a): 


.     (14) 


The  circumferential  motion  of  the  spring  will  be 

T=RT.  '  .  .  .'".v  ;?;;    .  (15) 


^  axial  extension  or  compression  of  the  spring  will 
be  found  by  the  aid  of  eqs.  (n)  and  (13)  : 


2 


-H.      (16) 


Eqs.  (6)  and  (7)  will  enable  any  spiral  spring  to  be 
designed  to  perform  a  given  duty  such  as  to  carry  a  pre- 
scribed load  or  serve  the  purposes  of  a  dynamometer,  while 
eqs.  (14)  and  (16)  will  give  the  distortions  of  the  spring, 
either  angular  or  axial. 

If  5  is  the  greatest  intensity  of  torsive  shear  in  a  normal 
section  of  the  spiral  rod  at  the  distance  r  from  the  centre, 
while  IP  is  the  polar  moment  of  inertia  of  the  section, 

Mt=s-IP.       y  .     ;    ...     (17) 
For  a  circular  section,  Ip=—. 

.    ,  2 

For  a  square  section,    Ip  =—(6  =  side  of  square). 
Eq.  (17)  gives: 


Art.  120.]  HELICAL  SPIRAL  SPRINGS.  757 

When  5  is  given, 


•-4 


2M 

1  (circular  section).     .     .     .     (i8a) 


ITS 


In  both  eqs.  (6)  and  (7),  MI  and  M  are  known  quanti- 
ties, as  they  are  the  given  loads. 

Again  if  k  is  the  intensity  of  stress  in  the  most  remote 
fibre  at  the  distance  d\  from  the  neutral  axis,  and  if  /  is 
the  moment  of  inertia  of  the  section  about  the  neutral  axis, 


......     (19) 

When  k  is  given, 


d\  =r=  A  H—T?  (circular  section).     .     .     (iga) 

\    irk 


The  two  intensities  5  and  k  exist  at  the  same  point, 
and  they  are  to  be  used  to  determine  the  greatest  intensities 
of  stress  in  the  cross-section  of  the  spiral  rod  precisely  as  was 
done  in  Art.  10. 

By  eq..'(2)  of  that  article,  the  greatest  and  least  inten- 
sities of  stress  (principal  stresses  of  opposite  kinds)  will  be  : 


k       I        k2 
ax.  intensity  =-  +\  s2  -\ —  (tension) ; 

2          \  A 


max 


k       I       k2 
min.  intensity  =-- \  s2-\ —  (compression). 

2      \         4 

At  the  opposite  end  of  that  diameter  of  section  of  the 
rod  normal  to  the  neutral  axis  where  k  is  compression, 
the  above  "  max.  intensity  "  will  be  compression  also, 
and  the  "  min.  intensity  "  will  be  tension. 


758  MISCELLANEOUS  SUBJECTS.  [Ch.  XVI. 

The  planes  on  which  these  principal  stresses  act  are 
given  by  eq.  (3)  of  Art.  (10) : 


25 

tan  2a=  — T-. 
k 


The  greatest  shear  at  the  same  point  is  given  by  eq. 
(6)  of  Art.  (9) ;  i.e., -its  intensity  is  half  the  difference  of  the 
principal  intensities,  or, 

_max.  —  min. 
2 

There  are  a  number  of  special  cases  which  may  easily 
be  developed  from  the  preceding  general  analysis. 

Small  Pitch  Angle. 

If  the  pitch  angle  <f>  is  so  small  that  sin  0  may  be  con- 
sidered zero  without  essential  error, 

sin  <£=o     and     cos  <£  =  i . 
Eqs.  (6)  and  (7)  then  give: 

(20) 


From  eqs.  (14)  and  (16): 

Ml 


PR2l  ' 

--'    '••••••     ('3) 


Art.  120.]  HELICAL  SPIRAL  SPRINGS.  759 

Rotation  of  Spring  Prevented. 

In  this  case  twisting  of  the  spring  is  prevented,  or  T  =o. 
Eq.  (14)  then  gives: 


Substituting  this  value  of  M  in  eq.  (16)  : 

_DP27(cos20  ,  sin20      (sine/)  cos  </>)2(Q'-Q)2  ]     f     . 

l  "' 


The  torsion  moment  Mt,  eq.  (6),  and  bending  moment 
Mb,  eq.  (7),  are  to  be  computed  by  using  the  value  of  M 
given  in  eq.  (24). 

Axial  Extension  or  Compression  Prevented. 
By  making  z  =o  in  eq.  (16), 


M          ^  sin  0  cos  0(Q'  -Q)  (  ,>. 

V  cos2  0+Q  sin  V     '     •     '     (26 


The  angle  of  twist  then  becomes: 

_        sin2  0     cos2  0       (sin  0  cos0)2(Q'  - 
"      fl  0'        ( 


For  circular  or  square  sections  Q'  —  Q  =  (  --  G'lf—  or  —  ) 

\2  /  \   2  67 

and  the  square  of  the  latter  alternative  factor  is  common 
to  (Qf  -QY  and  Q'Q  in  the  second  number  of  eq.  (27),  thus 
canceling  and  simplifying  the  numerical  application  of 
that  equation. 

In  computing  Mt  and  Mb,  eqs.  (6)  and  (7),    the  value 
of  Mi  given  by  eq.  (26)  is  to  be  used. 


760  MISCELLANEOUS  SUBJECTS.  [Ch.  XVI. 

This  form  of  helical  spring  is  employed  in  the  transmis- 
sion dynamometer. 

Work  Performed  in  Distorting  the  Spring. 

The  work  performed  in  producing  the  angular  and  axial 
distortions  r  and  z  by  the  moment  M  and  force  P  is  easily 
found  by  the  aid  of  eqs.  (14)  and  (16)  or  corresponding 
equations  for  special  cases.  The  couple  whose  moment 
is  M  performs  work  in  twisting  the  spring  through  the  arc 
T  (measured  at  unit  distance  from  the  axis  of  the  helix) 
expressed  by 


(28) 


The  force  P  performs  work  in  extending  or  compressing 
the  spring  the  distance  z  given  by  the  equation 


The  total  work  done  in  the  general  case  will  then  be: 

.   k     .     (30) 


corre 


For    special    cases,    as    already    indicated,    the 
sponding  value  of  T  and  z  must  be  used  in  eq.  (30). 

In  writing  the  preceding  equations  it  has  been  assumed 
that  both  M  and  P  are  gradually  applied.  If  they  were 
suddenly  applied,  the  distortions  would  be  2r  and  2Z  and 
oscillations  having  those  amplitudes  would  be  set  up. 
The  periods  of  the  amplitudes  would  depend  upon  the 
masses  moved. 


Art.  121.]  PLANE  SPIRAL  SPRINGS.  761 

Art.  121. — Plane  Spiral  Springs. 

A  plane  spiral  spring  may  be  represented  by  Fig.  i. 
The  outer  end  is  fastened  at  B,  but  the  inner  end  is  secured 
to  a  rotating  post  or  small  shaft  at  C.  The  spring  or  coil 
is  ' '  wound  up  "  to  an  increasing  number  of  turns  by  apply- 
ing a  couple  to  the  shaft  C,  as  in  winding  a  clock. 

As  a  couple  only  is  applied  at  C,  every  section  of  the 
spring  is  subjected  to  bending. by  the  same  couple,  i.e., 
there  is  a  uniform  bending  moment  throughout  the  entire 
spring.  This  uniform  condition  of  stress  makes  the  analysis 
of  this  spring  exceedingly  simple  if  the  thickness  of  the  metal 
is  small.  As  the  spring  is  a  spiral  beam  subjected  to  uni- 
form bending,  the  analysis,  to  be  perfectly  correct,  should 
be  based  on  that  for  curved  beams.  That  procedure  would 
introduce  much  complexity,  and  as  the  thickness  of  the  strip 
of  metal  constituting  the  spring  is  small  compared  with  its 
radius,  no  essential  error  is  committed  in  neglecting  the 
effects  of  curvature.  The  usual  cross-section  of  this  type 
of  spring  is  a  much  elongated  or  narrow  rectangle,  the 
greater  dimension  of  the  rectangle  being  parallel  to  the 
axis  of  the  couple  or  perpendicular 
to  the  plane  of  the  spring. 

If  u  is  the  unit  strain  at  unit 
distance  from  the  neutral  axis  of  a 
section  of  the  spring,  I  the  moment 
of  inertia  of  the  same  section  about 
the  neutral  axis,  and  E  the  modulus 
of  elasticity,  while  M  is  the  moment 
applied  at  C,  Fig.  i, 


M=EIu=  constant.    .     .     (i)  Fia  '• 

If  I  is  the  total  length  of  the  spring  and  0  the  total 
angular  distortion  for  that  length,   then  will  udl  be  the 


762  MISCELLANEOUS  SUBJECTS.  [Ch.  XVI. 

change  of  direction  or  angular  distortion  for  each  element 

dl.     Hence, 

Mdl=EIudl=EId(3.      ...          .     (2) 

Integrating  : 

and    fl,-..    .     .          .     (3) 


With  the  thin  metal  used  I  is  small  and  /3  may  be  a 
number  of  complete  circles,  perhaps  sufficient  to  wrap  the 
spring  closely  around  the  shaft  C. 

If  the  moment  M  is  applied  gradually,  the  work  done  in 
producing  the  total  angular  distortion  0  is 

,      M       M2l  ,  , 


This  is  the  same  as  the  expression  for  the  work  performed 
in  bending  "a  beam  by  a  moment  uniform  throughout  its 
length.  In  fact  the  plane  spiral  is  simply  a  special  case  of 
flexure,  the  bending  moment  being  uniform. 

If  the  moment  M  should  be  applied  suddenly,  the  total 
angular  distortion  would  be  2/3,  and  oscillations  having 
that  amplitude  might  be  set  up. 

Art.  122.  —  Problems. 

Problem  i.  —  A  helical  spring  having  a  diameter  of  helix 
of  3  inches  and  composed  of  twelve  complete  turns  of  a 
f-inch  round  steel  rod  sustains  an  axial  load  of  45  pounds. 
Find  the  axial  deflection  of  the  spring  and  the  greatest 
intensities  of  torsive  shear  and  bending  tension  and  com- 
pression in  the  rod. 

P=45lbs.;  #  =  1.5  ins.;     ^  =  15°;       £  =  117  ins.; 

E  =  30,000,000;       G  =  i2,  000,000;  r=i%in. 


Art.  122.1  PROBLEMS  FOR   ARTS.  120  AND  121.  763 

=  6S.5  in.-lbs.;  M=o; 


Q  =6^  =  23,373;  Q'  =£^  =  29,217. 

Substituting  these  quantities  in  eq.  (16): 

6faL 


23,373     29,217 
By  eqs.  (6)  and  (7)  : 

Mt=Mi  cos  </>  =66.  2  in.-lbs.; 
and 

M&=  —Mi  sin  <£=  —17.74  in.-lbs. 

Since  Ip=—  and  I=—  -,  eqs.  (18)  and  (19)  give: 
2  4 

5=9460  Ibs.  per  sq.in.  torsive  shear; 
^=3432  Ibs.  per  sq.in.  greatest  bending  stress. 

Problem  2.  —  Design  a  helical  spring  for  a  transmission 
dynamometer  for  8  H.P.  at  90  revolutions  per  minute. 
Axial  distortion  of  the  spring  is  prevented,  or  0=0.  Let 
low  working  stresses  and  other  data  be  taken  as  follows' 

k  =  16,000  Ibs.  per  sq.in.  ;  5  =  12,000  Ibs.  per  sq.in.  ; 

jR=3ins.;            </>  =  n°  /.    sin  </>  =  .191  and  cos<£  =  .982; 

(7  =  12,000,000;  £=30,000,000. 

MX9oX27r=8X33,ooo    .'.  M  =466.8  ft.  -Ibs.  =5602  in.-lbs. 

Q=G^    and       Q'=E^. 
2  4 

Eq.  (26)  then  gives: 

Mi  —  —  212  in.-lbs. 


764  MISCELLANEOUS  SUBJECTS.  [Ch.  XVI. 

By  eqs.  (6)  and  (7)  : 

in.-lbs.;     and     M6  =  554i  in.-lbs. 


Solving  eqs.  (18)  and  (19)  for  the  radius  of  the  rod:  ^ 
By  eq.  (i8a),  r  =  .$6  in.;     and  by  eq.  (iga),  r  =  .y6  in. 
Bending   of   the   rod,    therefore,    requires   the    greater 
radius,  and  r  =  .>]6  in.  will  be  taken. 

Eq.  (17)  gives  the  greatest  torsive  shear  in  a  section: 

s  =  1250  Ibs.  per  sq.in. 

The  equations  following  eq.  (19)  now  give: 

max.  intensity  =  +16,097  Ibs.  per  sq.in.; 
min.  intensity  =  —97  Ibs.  per  sq.in. 

The  spring  will  be  assumed  to  have  twelve  complete 
turns,  so  that  its  length  will  be: 

/  =  27r3  Xi2Xsec  0  =  230.  5  ins. 


The  twist  r  at  unit  distance  from  the  axis  of  the  helix 

now  becomes  : 

in. 


At  the  distance  of  10  inches  from  the  axis  the  twist 
would  be  1.59  inches,  but  the  spring  is  too  stiff  to  be  very 
sensitive.  A  higher  working  stress  k  may  properly  be  taken. 

If  in  the  same  problem  there  be  taken  120  revolutions 
per  minute  and  an  alloy  steel  for  which  the  working  stresses 
£=40,000  pounds  per  square  inch  and  5=30,000  pounds 
per  square  inch  may  be  prescribed,  then  by  using  the 
results  already  established: 

M  —-^—  Xs6o2  =4200  in.-lbs.; 
120 


Art.  122.]  PROBLEMS  FOR  ARTS.  120  AND  121.  765 

M\  =  —  f  X  2  1  2  =  —  159  in.-lbs.  ; 
M<  =  1x862=647  in.-lbs.; 
^&  =  fX554i  =4156  in.-lbs. 

3  1?          j 

For  shear:  r  =^p  X—  X-36  =.67  X-36  =.24  in. 

\4      2-5 

3/~       ~ 
For  bending:  r=\^-X  —  X.  76  =.67  X.  76  =.51  in. 

M      2.5 

.150 


At  the  distance  of  10  inches  from  the  axis  of  the  helix 
the  twist  would  be  ioX.  795  =  7.  95  inches. 

Problem  3.  —  What  will  be  the  angular  distortion  /3 
of  a  plane  spiral  spring  i  inch  by  ^¥  inch  in  section  and 
20  inches  long  if  the  distorting  moment  is  10  inch-pounds. 
Eq.  (3)  of  Art.  121  gives: 

10X20  10X20X12X125,000 

P    =  -  ^L   =  -     =   IO 
3O,OOO,OOOX/  30,000,000 

(about  i^  complete  turns). 
The  fibre  stress  is 

ioX— 

,  100  1U 

k  =  -  =150,000  Ibs.  per  sq.m. 


12  X  125,000 

Art.  123. — Flat  Plates. 

The  correct  analysis  of  stresses  in  loaded  flat  plates  even 
of  the  simplest  form  of  outline  has  not  yet  been  made  suf- 
ficiently workable  for  ordinary  engineering  purposes,  either 
for  plates  simply  resting  on  edge  supports  or  with  edges  of 
plates  rigidly  fixed  to  their  supports.  It  is  necessary, 


766  MISCELLANEOUS  SUBJECTS.  fCh.  XVI. 

therefore,  to  combine  simple,  but  approximate  analysis 
based  on  reasonable  assumptions,  with  experimental  results 
so  as  to  obtain  workable  formulae.  The  following  pro- 
cedures, due  chiefly  to  Bach  and  Grashof,  are  commonly 
employed  in  treating  flat  plates: 

Square  Plates  —  Uniform  Load. 

In  Fig.   i  let  ABCD  represent  a  square  plate  simply 
resting  on  the  edges  of  a  square  opening.     Tests  of  such 
/    plates  by  Bach  have  shown  that  when 
TB      increasingly   loaded    they   will    ulti- 
mately fail  along  a  diagonal,  as  AB. 
Let  the  plate  be  uniformly  loaded 
with  p  pounds  per  square  unit,  then 
let  moments  be  taken  about  the  diag- 
onal AB.      If  b  is  the  side  of   the 
square,    the  load  on  the  triangular 

half  of  the  square  is  £—  ,  and  the  distance  of  its  centre 

2 

from  AB  is  Jfe  sin  45°  =.2366.  The  upward  supporting 
forces  or  reaction  on  the  sides  AD  and  DB  will  also  be 

half  the  load  on  the  plate,  £-,  and  its  centre  will  be  at  the 

2 

distance  -  ^^-=,3S4b  from  AB  Hence  the  moment 
about  A  B  will  be: 

~-  =.059^3.     .     .     .     (i) 


If  h  be  the  thickness  of  the  plate,  the  moment  of  inertia 
/  about  its  neutral  axis  will  be: 

T     b  sec  45°  h3 


Art.  123.]  SQUARE  PLATES.  767 

The  ordinary  flexure  formula  then  gives  for  the  greatest 
intensity  of  bending  stress  k,  assuming  it  to  be  uniform 
throughout  the  diagonal  section, 


,, 

M- 


Or,'  if  the  thickness  is  desired, 

.......      (4) 


Eq.  (4)  gives  the  thickness  of  plate  required  to  carry 
the  unit  load  p  when  the  working  stress  is  k. 

Square  Plates  —  Single  Centre  Load. 

If  a  single  load  P  rests  at  the  centre  of  a  square  plate, 
using  Fig.  i  and  following  the  same  method  as  in  the 
preceding  section,  the  moment  about  the  diagonal  AB 
will  be  : 

,.,     Pb  sin  45°  „,  ,  . 

M=  --  -=.i77P6  .....     (5) 

The  moment  of  inertia  I  is  the  same  as  befoie  and  it 
is  given  by  eq.  (2).  Hence,  assuming  a  uniform  intensity 
k  throughout  the  extreme  fibres  : 


(6) 


a! 

Or, 

fo 

(7) 


768 


MISCELLANEOUS  SUBJECTS, 


[Ch.  XVI. 


Rectangular  Plates  —  Uniform  Load. 

Fig.  2  shows  a  rectangular  plate  with  sides  a  and  b. 
With  a  much  oblong  rectangle  the  indications  of  tests  are 
not  so  well  defined  as  to  the  section  of  failure,  but  tenta- 
tively the  diagonal  section  AB  may  be  taken  as  a  close 
approximation  for  usual  proportions.  DF  is  a  normal 
to  A  B  drawn  from  D.  The  uniform  load  on  the  triangular 


half  ABD  of  the  plate  is      —  and  its  centre  of  action  is  at 

2 

the  normal  distance  -  from  AB.     The  centre  of  the  sup- 

o 

porting  forces  or  reaction  along  the  edges  AD  and  DB  is 

-  from  A  B.     Hence  the  moment  about  A  B  is 
2 


M 


_pabin     n\  _pabn 


2     \2 


12 


•     .     (8) 


n\ 


FIG.  2. 
Referring  to  Fig.  (2) : 

n=b  sin  </>    and     AB  =b  sec  $.      .     .     (9) 
Therefore  the  moment  of  inertia  of  the  diagonal  section 

sin  0     Jb  sec  cf>h2 


is 


T     b  sec  4>h?  , 

/= — -;     and 


12 


12 


Art.  123.] 
Hence, 

k=p 


CIRCULAR  PLATES. 


769 


ab  sin  0  cos  0 


or        = 


sn 


cos  ^.    (10) 


As  is  obvious,  P  =pab  is  the  total  load  on  the  plate. 

Rectangular  Plate — Centre  Load. 

If  a  single  load  P  rests  at  the  centre  of  the  plate,  the 
moment  about  the  diagonal  AB,  Fig.  2,  is  produced  by  the 
reaction,  only,  of  tKe  supporting  forces  along  the  edges 
AB  and  BD,  and  its  value  is  * 

M=    n  _^Pb  sin 

2  2 

Consequently, 
,      $P  sin  0  cos 


or, 


....  (n) 

sin  0  cos  0.       (12) 


Circular  Plate — Uniform  Load — Centre  Load. 

The  circular  plate  with  radius  r  is  shown  in  Fig.  3. 
The  same  general  assumptions  are  made  as  in  the  preceding 
cases,  i.e.,  uniform  condition  of 
bending  stress  throughout  the 
section  of  failure  and  uniform 
support  along  the  edge  of  the 
plate.  It  is  clear  that  the  latter 
assumption  is  strictly  correct  for 
the  circular  outline.  Any  diam- 
eter, as  AB,  may  be  taken  for 
the  section  of  failure. 

It  will  be  convenient  to  sup- 
pose the  uniform  load  to  be  ap- 
plied on  a  circle  of  radius  n,  as  shown  in  Fig.  3.  Then 

TTfl2 

the  load  on  half  of  the  plate  is  p and  its  centre  is  at 


770  MISCELLANEOUS  SUBJECTS.  [Ch.  XVI. 

the  distance  —  from  AB.     The  edge-supporting  force  or 

.3*- 
reaction,  equal  to  the  half  load  on  the  plate,  has  its  centre 

2  Y 

at  the  distance  of  —  from  AB.     The  moment  about  the 

7T 

latter  diameter  is,  therefore, 


If  h  is  the  thickness  of  the  plate,  as  in  the  preceding 
cases,  the  moment  of  inertia  I  is 


j-_*>">  _rh3 
12    "IT 


Hence, 

•/''"    ,  *=M2^-T^ (16) 

Or, 


If  the  load  is  uniform  over  the  entire  circular  plate, 
r=ri,  and 


and,     h  =  r.      .     .     (18) 


If  the  load  is  concentrated  at  essentially  a  point,  r\  =o, 
but        l   must  be  displaced  by  —  ; 


2  2 


7T/? 

These    formulae    for    circular   plates    are    more    nearly 


;    and,    h-.      .    .     (19) 

7T/? 


Art.  123.] 


ELLIPTICAL    PLATES. 


771 


correct  in  analysis  and  give  results  more  nearly  in  agree- 
ment with  tests  than  those  derived  for  other  cases. 

Elliptical  Plates  —  Centre  Load  —  Uniform  Load. 

An  elliptical  plate  is  shown  in  Fig.  4.  The  approximate 
formulae  for  this  case  may  be  conveniently  established  by 
first  considering  two  axial  strips  of 
the  same  (unit)  width,  the  length 
of  A  B  being  2  a  and  of  CD,  26,  a 
single  load  being  placed  at  their 
intersection.  The  centre  deflec- 
tions of  the  two  strips  as  parts  of 
the  plates  must  be  the  same.  Let 
Pi  be  the  centre  load  for  the  strip 
AB,  and  P2  the  centre  load  for  CD. 

The  desired  centre  deflection  for  each  strip  acting  as  a  beam 
is  given  by  eq.  (28),  Art.  28.  The  equality  of  the  two  de- 
flections gives  the  equation,  2  a  being  one  span  and  2b  the 
other  : 


FIG.  4. 


P2     as 

W 


(20) 


ness 


6EI      6EI  ' 

h? 
As  each  strip  .is  of  unit  width,  /  =  —  ,h  being  the  thick- 

12 

of  plate.     Hence  the  greatest  fibre  stresses  are 
Mh        ^  a  b 


=3Pi:     and, 


,     , 
(21) 


Eqs.  (21)  and  (20)  then  give: 

ki=Pia 
k2     P2  b 


b2 


(22) 


Eq.  (22)  shows  that  k2  is  the  greatest  fibre  stress  and, 
hence,  that  the  major  axis  of  the  ellipse  will  be  the  line  of 
failure,  as  would  be  anticipated  without  the  analysis. 


772  MISCELLANEOUS  SUBJECTS.  [Ch.  XVI. 

.  If  the  ellipse  of  Fig.  4  be  elongated  by  lengthening 
the  major  axis  2  a  to  infinity,  the  result  will  be  a  corre- 
spondingly long  rectangular  plate  of  26  width  or  span. 
Hence,  the  greatest  fibre  stress  for  this  case  of  uniform 
load  will  be  for  a  unit  cross  strip  of  plate: 

Mh_p(2b)2    hl2 

=  2!  '        8    "afca 

This  is  the  greatest  intensity  of  stress  for  an  ellipse 
whose  major  axis  2  a  is  infinity.  The  other  extreme  is  the 
circle  for  which  the  greatest  intensity  of  stress  is,  eq.  (18), 


For  ellipses  in  general,  in  the  absence  of  a  satisfactory 
analysis,  it  is  tentatively  proposed  to  write: 


When  b=  a,  eq.  (25)  gives  the  correct  value  for  a  circle, 
and  when  b=o  the  result  is  correct  for  the  extreme  ellipse. 
The  thickness  of  plate  for  a  given  uniform  load  p  is 


(26) 


Flat  Plates  Fixed  at  Edges. 


Grashof  and  others  have  partly  by  analysis  and  partly 
empirically  deduced  a  number  of  formulae  for  plates  fixed 
at  their  edges,  i.e.,  encastre,  instead  of  simply  supported. 
The  following  have  been  used  and  may  be  considered  fairly 
satisfactory,  using  the  same  notation  as  in  the  preceding 
parts  of  this  article. 

I.  Circular  plate  with  radius  r  and  uniform  load  p. 
The  greatest  intensity  of  stress  is,  if  h  is  the  thickness, 


Art.  123.]  PLATES  WITH  FIXED  EDGES.  773 

.         and>  /t 


II.  Stayed  flat  surfaces,  stay  bolts  being  the  distance 
c  apart  in  two  directions  at  right  angles  to  each  other. 
Each  stay  carries  the  uniforni  load  pc2.  The  greatest 
intensity  of  stress  may  be  taken: 


:    and,     h=-.  •,,,,.     (28) 


III.  Rectangular  plate  a  long,  b  wide,  supporting  uniform 
unit  load  p.     The  greatest  intensity  of  stress  may  be  taken  : 


a4        b2  ,      7        9,     p        i  ,     x 

;    and'    fc=0*-     (29) 


If  the  plate  is  square,  a  =  b  : 

k2  j      /  f     \ 

k=P-~:     and,     h=-. 

All  these  plates  with  edges  either  fixed  or  simply  sup- 
ported are  supposed  to  be  truly  flat,  as  any  arching  or 
dishing  changes  materially  the  conditions  of  stress. 

Problem  i.  —  What  thickness  of  steel  plate  is  required 
to  carry  a  load  of  200  pounds  per  square  inch  over  a  rect- 
angular opening  24  by  36  inches.  Eq.  (10)  gives  the 
expression  for  the  thickness  h  of  the  plate  when  simply 
supported  along  its  edges.  The  total  load  isP  =  2ooX36X 
24  =  168,800  pounds. 

tan  </>=f!  =.667    /.    4>  =33°  40'  and  sin  0  cos  0  =  .  461. 
If  the  working  stress  &==  16,000  pounds  per  square  inch; 

x.  46  1  =1.56  inches.     A    plate     i  A-  inches 


2  X  16,000 
thick,  therefore,  meets  the  requirements. 

Problem  2.  —  Design  a  circular  steel  plate,  simply  sup- 
ported on  its  edge,  for  an  opening  30  inches  in  diameter 


774  MISCELLANEOUS  SUBJECTS.  [Ch.  XVI. 

to  carry  a  load  of  100  pounds  per  square  inch,  if  k  =  15,000 
pounds  per  square  inch.  r  =  i5  inches  and  P  =  iooXirr2  = 
100  X  706.9  =  70,690  pounds.  _ 

Eq.    (18)    then    gives:     h  =  i$<J  I0°  =1.22  inches.    A 


plate  i  \  inches  thick  will  therefore  be  satisfactory. 

If  the  plate  were  rigidly  fixed  along  its  edge^eq.  (27) 
shows  that  the  thickness  would  be:  h  =  i.22V%  =  i  inch 
thick. 

Art.  124.    Resistance  of  Flues  to  Collapse. 

IF  a  circular  tube  or  flue  be  subjected  to  external  normal 
pressure,  such  as  that  of  steam  or  water,  the  material  of 
which  it  is  made  will  be  subjected  to  compression  around 
the  tube,  in  a  plane  normal  to  its  axis.  If  the  following 
notation  be  adopted, 

/  =  length  of  tube; 
d  =  diameter  of  tube  ; 

t  =  thickness  of  wall  of  the  tube  ; 
p  =  intensity  of  excess  of  external  pressure  over  internal  ; 

then  will  any  longitudinal  section  It,  of  one  side  of  the  tube, 
be  subjected  to  the  pressure  —  .  But  let  a  unit  only  of 

length  of  tube  be  considered.  This  portion  of  the  tube  is 
approximately  in  the  condition  of  a  column  whose  length 
and  cross-section,  respectively,  are  xd  and  t. 

The  ultimate  resistance  of  such  a  column  is  (Art.  35) 


As  this  ideal  column  is  of  rectangular  section, 
r_£ 

12' 


Art.  124.]  RESISTANCE  OF  FLUES  TO  COLLAPSE.  775 

and 

Ft3 

p-:=— 
~2' 


But  P  =  pd,  hence 

Et3 


is  the  greatest  intensity  of  external  pressure  which  the  tube 
can  carry.  But  the  formulae  of  Art.  35  are  not  strictly 
applicable  to  this  ideal  column.  The  curvature  on  the  one 
hand  and  the  pressure  on  the  other  tend  to  keep  it  in  position 
long  after  it  would  fail  as  a  column  without  lateral  support. 
Hence  p  will  vary  inversely  as.  some  power  of  d  much  less  than 
the  third. 

Again,  it  is  clear  that  a  very  long  tube  will  be  much  more 
apt  to  collapse  at  its  middle  portion  than  a  short  one,  as  the 
latter  will  derive  more  support  from  the  end  attachments; 
and  this  result  has  been  established  by  many  experiments. 
Hence  p  must  be  considered  as  some  inverse  function  of  the 
length  /. 

Eq.  (i),  therefore,  can  only  be  taken  as  typical  in  form, 
and  as  showing  in  a  general  way,  only,  how  the  variable 
elements  enter  the  value  of  p.  If  x,  y,  and  z,  therefore,  are 
variable  exponents  to  be  determined  by  experiment,  there 
may  be  written 


in  which  c  is  an  empirical  coefficient. 

Sir  Wm.  Fairbairn  ("Useful  Information  for  Engineers, 
Second  Series")  made  many  experiments  on  wrought-iron 
tubes  with  lap-  and  butt-joints  single  riveted.  He  inferred 


776  MISCELLANEOUS  SUBJECTS.  [Ch.  XVI. 

from  his  tests  that  y  =  z  =  i.     Two  different  experiments 
would  then  give 

pld  =  ct*,      .......     (3) 

p'l'd'=ct'x  ........     (4) 

Hence 

log  (pld)=log 


in  which  "log"  means  "logarithm."     Subtracting  one  of 
these  last  equations  from  the  other,  the  value  of  x  becomes 


log(pld)-log(p'l'd') 

log,_logr        -  .-.     (5) 


As  p,  I,  d,  t,  p',l',d',  and  t'  are  known  numerical  quantities 
in  every  pair  of  tests,  x  can  at  once  be  computed  by  eq.  (5)  ; 
c  then  immediately  results  from  either  eq.  (3)  or  eq.  (4). 
By  the:  application  of  these  equations  to  his  experimental 
data,  Fairbairn  found  for  wrought-iron  tubes: 

,  .  t2-19 

f  ^  =  9,675,  600-^-,  ......     (6) 

in  which  p  is  in  pounds  per  square  inch,  while  t,  /,  and  dare 
in  inches.  Eq.  (6)  is  only  to  be  applied  to  lengths  between  18 
and  120  inches. 

He  also  found  that  the  following  formula  gave  results 
agreeing  more  nearly  with  those  of  experiment,  though  it  is 
less  simple: 

•  t2-19  d 

£  =  9,675,600-^-0.002^.      ....     (7) 


Art.  124.]  RESISTANCE  OF  FLUES  TO  COLLAPSE.  777 

Fairbairn  found  that  by  encircling  the  tubes  with  stiff 
rings  he  increased  their  resistance  to  collapse.  In  eases 
where  suck  rings  exist,  it  is  only  necessary  to  take  for  I  the 
distance  between  two  adjacent  ones. 

In  1875  Prof.  Unwin,  who  was  Fairbairn'  s  assistant  in 
his  experimental  work,  established  formulae  with  other 
exponents  and  coefficients  ("  Proc.  Inst.  of  Civ.  Engrs.," 
Vol.  XLVI).  He  considered  x,  y,  and  z  variable,  and 
found  for  tubes  with  a  longitudinal  lap-joint: 

t2-1 
£  =  7»363,<>oo/^lT6-    .....     (8) 

From  one  tube  with  a  longitudinal  butt-joint,  he  deduced: 

t2-21 
£  =  9,614,000^^.   .....     (9) 

For  five  tubes  with  longitudinal  and  circumferential  joints, 
he  found: 

£2.35 

£  =  15,547,000^^.     ....     (10) 

By  using  these  same  experiments  of  Fairbairn,  other 
writers  have  deduced  other  formulas,  which,  however,  are 
of  the  same  general  form  as  those  given  above.  It  is  proba- 
ble that  the  following,  which  was  deduced  by  J.  W.  Nystrom, 
will  give  more  satisfactory  results  than  any  other: 


At  the  same  time,  it  has  the  great  merit  of  more  simple 
application. 

From  one  experiment  on  an  elliptical  tube,  by  Fairbairn, 
it  would  appear  that  the  formulae  just  given  can  be  approxi- 


778 


MISCELLANEOUS  SUBJECTS. 


[Ch.  XVI. 


mately  applied  to  such  tubes  by  substituting  for  d  twice 
the  radius  of  curvature  of  the  elliptical  section  at  either 
extremity  of  the  smaller  axis.  If  the  greater  diameter  or 
axis  of  the  ellipse  is  a  and  the  less  6,  then,  for  d,  there  is 

to  be  substituted  -r-. 
6 


Art.  125.  —Approximate  Treatment  of  Solid  Metallic  Rollers. 

An  approximate  expression  for  the  resistance  of  a  roller 
may  easily  be  written.  The  approximation  may  be  con- 
sidered a  loose  one,  but  it  furnishes  a  basis  for  an  accurate 
empirical  formula. 

The  following  investigation  contains  the  improvements 
by  Prof.  J.  B.  Johnson  and  Prof.  H.  T.  Eddy  on  the 

method  originally  given  by  the 
author. 

The  roller  will  be  assumed 
to  be  composed  of  indefinitely 
thin  vertical  slices  parallel  to 
its  axis.  It  will  also  be  as- 
sumed that  the  layers  or  slices 
act  independently  of  each 
other. 

Let  E'  be  the  coefficient  of 
elasticity  of  the  metal  over  the 
Fro.  i  roller. 

Let  E  be  the  coefficient  of  elasticity  of  the  metal  of 
the  roller. 

Let  R  be  the  radius  of  the  roller  and  Rf  the  thickness 
of  the  metal  above  it. 

Let  w  =  intensity  of  pressure  at  A  ; 

p=  "  any  other  point; 


Art.  125.]     TREATMENT  OF  SOLID  METALLIC  ROLLERS.  779 

Let        P=  total  weight  which  the  roller  sustains  per  unit 

of  length. 

x  be  measured  horizontally  from  A  as  the  origin  ; 
d=AC; 
e=DC. 

From  Fig.  i  : 

wR-     A'W     pR 
--,    AB  =--. 


+        .    .    .    (i) 
and 

A'C'-A'B'+B'V-p+.     ...    (2) 

Dividing  eq.  (2)  by  eq.  (i), 


But 


If  the  curve  DAH  be  assumed  to  be  a  parabola,  as  may 
be  done  without  essential  error,  there  will  result: 

/-\-  e  A 

=  3*  * 
Hence 

(3) 


?8o 
But 


MISCELLANEOUS  SUBJECTS. 


[Ch.  XVI. 


e  =  \/2Rd  —  d2  =  \/2Rd,  nearly. 


By  inserting  the  value  of  d  from  eq.  (i)  in  the  value  of 
?,  just  determined,  then  placing  the  result  in  eq.  (3), 


p  =—' 


(4) 


p=4-l 


EE' 


(5) 


The  preceding  expressions  are  for  one  unit  of  length. 
If  the  length  of  the  roller  is  /,  its  total  resistance  is 


Or  if  R=R', 


R    R 


(6) 


(7) 


In  ordinary  bridge  practice  eq.  (7)  is  sufficiently  near 
for  all  cases. 

A  simple  expression  for  conical  rollers  may  be  obtained 
by  using  eqs.  (4)  or  (5). 

I* i     3! 

r  71 

|«---* * 


TIG.     2 

As  shown  in  Fig.  2,  let  z  be  the  distance,  parallel  to  the 
axis,  of  any  section  from  the  apex  of  the  cone ;  then  con- 


Art.  126.]     RESISTANCE   TO  DRIVING  AND  DRAWING  SPIKES.       781 

sider  a  portion  of  the  conical  roller  whose  length  is  dz.  Let 
Rl  be  the  radius  of  the  base.  The  radius  of  the  section 
under  consideration  will  then  be 


and  the  weight  it  will  sustain,  if  R1=R\ 


Hence 


-/' 


Eqs.  (6),  (7),  and  (8)  give  ultimate  resistances  if  w  is 
the  ultimate  intensity  of  resistance  for  the  roller. 

It  is  to  be  observed  that  the  main  assumptions  on  which 
the  investigation  is  based  lead  to  an  error  on  the  side  of 
safety. 

If  for  wrought  iron,  w  =  12,000  pounds  per  square  inch, 
and  £  =  £'  =  28,000,000  pounds,  eq.  (5)  gives 


Art.  126.  —Resistance  to  Driving  and  Drawing  Spikes. 

Some  very  interesting  experiments  on  driving  and  draw- 
ing rail  spikes  were  made  by  Mr.  A.  M.  Wellington,  C.E., 
and  reported  by  him  in  the  "  R.  R.  Gazette,"  Dec.  17,  1880. 
He  experimented  with  wood  both  in  the  natural  state  and 
after  it  had  been  treated  by  the  Thilmeny  (sulphate  of 
baryta)  preserving  process. 

11  The  test-blocks  were  reduced  to  a  uniform  thickness  of 
4.5  inches,  this  thickness  being  just  sufficient  to  give  a  full 


782 


MISCELLANEOUS  SUBJECTS. 


[Ch.  XVI. 


bearing  surface  to  the  parallel  sides  of  the  spikes  when 
driven  to  the  usual  depth,  and  to  allow  the  point  of  the 
spike  to  project  outwards.  It  was  considered  that  the  be v- 

TABLE  I. 
SPIKES  WERE  STANDARD:   5.5  INCHES  X  T9ir  INCH. 


Kind  of  Wood. 

-      Natural  Wood. 

Prepared  Wood. 

To  Driving 
Spike,  Pounds. 

To  Pulling 
Spike,  Pounds. 

To  Driving 
Spike,  Pounds. 

To  Pulling 
Spike,  Pounds. 

Beech  ...            

I  5,  216 
(  6,743 
,  5,970 
}  5,670 
5,  216 
5,521 

J  6,433 
1  6,433 

3,996 

4,202 

4,453 
4,758 
3,996 
3,386 
4,148 
3,538 
j  4,103 
I  3,493 

Mean. 
5,98o 

•5,820 
•5,368 
5,953 
•6,433 

4,09° 

4,606 
3,691 
3,843 

3,798 
2,910 

Mean. 

!:&}'•"» 
l:lll\^ 

6$K'» 

4,560 

!&V*' 
ifiSK*' 

»•«- 
S5  ** 

aal.*-» 

»"« 

1,996 

Mean. 

?»<*« 

«:l'sl\  5,353 

!&[M* 

Mean. 
SS?K" 

6:$h«» 

(Split) 

White  oak  green   

Pin  oak  

White  ash  

White  oak,  well  seasoned.  .  . 
Black  ash  

4,4531 

asM 

3,38oJ 

tai**« 

3.340! 

»•- 

3.493J 

j:£K-» 

Elm     

Soft  maple  

l^lh^ 
&&& 

J:!!i}'-8» 

1,968 

Sycamore  

Hemlock 

elled  point  could  add  very  little  to  the  holding  power  of  the 
spike,  and  it  was  desired  to  press  the  spike  out  again  by 
direct  pressure  after  turning  the  block  over.  ..." 

The  forces  exerted  in  pulling  and  driving  the  spikes 
were  produced  by  a  lever.  A  few  tests  with  a  hydraulic 
press  showed  that  the  friction  of  the  plunger  varied  from 
about  6  to  1 8  per  cent.  The  experimental  results  are 
given  in  Table  I. 

Some  very  excellent  tests  of  the  holding  power  of  rail- 
road spikes  and  lag-screws  were  made  by  Mr.  A.  J.  Cox,  of 
the  University  of  Iowa,  during  1891,  in  the  engineering 
laboratory  of  that  institution,  the  results  of  which  were 


Art.  126.]  RESISTANCE  TO  DRIVING  AND  DRAWING  SPIKES.          783 


TABLE  II. 

RESISTANCE  OF  RAILROAD  SPIKES  TO  PULLING  OUT  AND 
PRESSING  IN. 


Kind  of  Tie  and  Spike. 

No. 
of 
Tests. 

Greatest  Resistance 
in  Pounds. 

Average 
Resistance 
in  Pounds 
per  Square 
Inch  Sur- 
face of  Spiket 

Average 
Resistance 
per  Ounce 
of  Spike. 

Maxi- 
mum. 

Average. 

Mini- 
mum. 

Seasoned  White-oak  Tie. 

Common  spike  
Common  spike,  £-in.  bored  hole. 

20 
9 
I 

2 

4 
3 

7 
3 
5 
3 

2 
2 
2 
2 

7,700 
6,660 

5,5i4 
,  4,936 

1    5,120* 

3,5oo 
3,950 

643 

575 

520 
378 

S48 
716 

133 
162 

664 
595 

537 

390 

632 
934 

567 
740 
555 
784 

137 
169 
192 
140 

Common  spike,  £-in  bored  hole,  f 

6,580 
6,850 

6,130 
5,950 
5,68o 
6,930 

i  ,240 
1,460 
1,830 
980 

1  4,460   • 
\  4,040* 

\ 
f 

\  3,240 
5,843 
6,350 

4,706 

5,807 

5,130 

5,334 

1,140 
1,400 
1,775 
955 

i 
5,290 
5,600 

4,050 
5,720 
4,400 
4,030 

1,040 
1,340 
1,720 
930 

Hill  curved  spike    . 

Bayonet  spike  
Unseasoned  White  Oak. 
Common  spike.  .  . 

Common  spike,  £-in.  bored  hole. 
Hill  curved  spike 

Bayonet  spike  

Unseasoned  White  Cedar. 
Common  spike. 

Common  spike,  £-in.  bored  hole. 
Hill  curved  spike  

Bayonet  spike  

PRESSING  SPIKES  INTO  TIES  UNDER  STEADY  PRESSURE  OF 
TESTING  MACHINE. 


White-oak  Ties. 

Curved  spike,  pulling   out  
Bayonet  spike,  pressing  in  

2 
2 

7,43° 
6,830 
6,660 

7,375 
6,615 
6,530 

7,320 
6,400 
6,400 





Bayonet  spike,  pulling  out  

2 

4,400 

3,845 

3,290 

*  These  values  are  the  first  resistance  to  drawing  out. 
in  the  same  holes  and  redrawn,  with  the  results  shown, 
t  Wedge  surface  not  considered. 


The  spikes  were  then  redriven 


published  in  the  technical  journal  ("The  Transit")  of  the 
university  for  September,  1891;  they  will  be  found  some- 
what rearranged  in  Tables  II  and  III.  Three  kinds  of 
spikes  were  used,  viz.,  the  common  spike  (length  5.5  ins., 
0.5625  in.  square,  weight  8.3  oz.),  Hill's  curved  spike  (length 
5.875  ins.,  weight  9.25  oz.),  and  the  bayonet  or  grooved 


784 


MISCELLANEOUS  SUBJECTS. 


[Ch.  XVI, 


spike  (length  5.5  ins.,  weight  6.8  oz.).     The  timber  of  the 
ties  is  shown  in  the  two  tables.      The  spikes  were  forced 

TABLE  III. 
RESISTANCE  OF  LAG-SCREWS  TO  PULLING  OUT.* 


Diameter 

Diameter 

Length 

Maximum 

Resistance 

No 

Kind  of  Wood. 

of 
Screw, 

of  Bored 
Hole, 

Screw 
in  Hole, 

Average 
Resistance 

pounds 
per  Square 

of 

Inches. 

Inches. 

Ins. 

in  Pounds. 

Inch. 

Seasoned  white  oak.  .  .  . 
Seasoned  white  oak.  .  .  . 

I 

£ 

4V2 

3 

8,037 
6,480 

1,024 
1,223 

3 

i 

Seasoned  white  oak.  .  .  . 

¥2 

% 

4^2 

8,780 

1  ,23Q 

2 

Yellow-pine  stick  

% 

¥2 

4 

3,800 

484 

2 

White  cedar  unseasoned 

5/8 

% 

4 

3,405 

434 

2 

*  The  area  of  surface  for  these  lag-screws  used  in  finding  the  resistance  per  square  inch 
was  computed  as  that  of  a  cylinder  whose  diameter  was  equal  to  the  diameter  of  the  screw 
considered.  In  pulling  the  first  lag-screw  of  Table  III,  the  resistance  of  8037  pounds  at 
the  end  of  a  i-inch  movement  decreased  to  4550,  2476,  1475,  and  410  pounds  at  the  ends 
of  movements  of  0.5,  i,  2,  and  2.75  inches  respectively. 

into  the  wood  by  the  pressure  exerted  by  the  ioo,ooo-pound 
testing  machine  used  in  the  tests,  and  by  which  they  were 
pulled  out  of  the  ties. 

The  greatest  pulling  resistance  of  any  spike  is  offered 
at  the  very  beginning  of  motion,  and  it  then  rapidly  de- 
creases. A  common  spike  which  resisted  5120  pounds  at 
the  beginning  of  motion  offered  but  3050  pounds  after 
having  moved  a  half-inch,  2,440  pounds  after  i  inch  of  mo- 
tion, 1,300  pounds  after  1.75  inches,  940  pounds  after  2 
inches,  and  440  pounds  after  moving  3  inches;  the  original 
penetration  of  the  spike  was  4.375  inches  in  a  seasoned  white- 
oak  tie.  Similar  results  were  reached  with  other  timbers. 

When  spikes  were  pressed  into  the  ties  the  timber 
offered  an  increasing  resistance  to  penetration,  but  at  a 
rate  less  rapid  than  that  of  the  decrease  in  pulling  out.  A 
^-inch  penetration  in  a  seasoned  white-oak  tie  gave  a  re- 
sistance to  a  common  spike  of  2,320  pounds  which  increased 
to  3,340  pounds  for  i-inch  penetration,  to  4550  pounds  for 


Art.  126.]  RESISTANCE  TO  DRIVING  AND  DRAWING  SPIKES.          785 

2  inches,  to  5580  pounds  for  3.5  inches,  and  to  6555  pounds 
for  4.5  inches. 

The  following  results  showing  the  relative  holding 
power  of  common  and  screw  railroad  spikes  were  found 
by  tests  made  by  Prof.  W.  Kendrick  Hatt  for  the  U.  S. 
Dept.  of  Agriculture  and  published  in  Forest  Service 
Circular  46,  1906. 


TABLE  IV. 

HOLDING  FORCE  OF  COMMON  AND  SCREW  SPIKES. 


Species  of  Wood  and 
Kind  of  Spike. 

Num- 
ber of 
Tests. 

Condition  of  Wood. 

Force  Required  to  Pull  Spike. 

Average. 

Max. 

Min. 

White  oak: 
Common  spike  
Screw  spike  
Ratio  

5 
5 

Partially  seasoned.  . 
Partially  seasoned.  . 

Pounds. 
6,950 
13,026 
1.88 

Pounds. 
7,870 
14,940 

Pounds. 
6,  1  60 
11,050 

Oak  (probably  red)  : 
Common  spike  
Screw  spike  
Ratio  

5 
8 

Seasoned  

4,342 
11,240 
2.61 

5,300 
13,530 

3,490 
8,900 

Seasoned  

Loblolly  pine: 
Common  spike  
Screw  spike. 

28 
26 

Seasoned  
Seasoned 

3,670 

7,748 

2.  II 

6,000 
14,680 

2,320 
4,170 

Ratio  

Hardy  catalpa: 
Common  spike  
Screw  spike  

12 
14 

Green 

3,224 
8,26l 
2.56 

4,000 
9,440 

2,190 
6,280 

Green. 

Ratio  

Common  catalpa: 
Common  spike  
Screw  spike  

II 
II 

Green.    .  . 

2,887 

6,939 
2.42 

4,500 
8,340 

2,240 
5,890 

Green  

Ratio  

Chestnut: 
Common  spike  
Screw  spike  

4 

5 

Seasoned  

2,980 
9,418 

7.  JC 

3,220 

11,150 

2,600 
7,470 

Seasoned 

Ratio  

786 


MISCELLANEOUS  SUBJECTS. 


[Ch.  XVI. 


TABLE  V. 

HOLDING  FORCE  OF  COMMON  AND  SCREW  SPIKES. 
SEASONED  CLEAR  AND  KNOTTY  LOBLOLLY  PINE  TIES. 


Position  of  Spike. 

Kind  of  Spike. 

Number 
of  Tests. 

Force  Required  to  Pull  Spike, 

Average. 

Max. 

Min. 

In  clear  wood 

Common  

36 
18 
40 
20 

Pounds. 
3,466 
2,615 

7,180 
9,763 

Pounds. 
6,250 
3,750 
I3,7io 
17,200 

Pounds. 
i,  880 

1,010 
2,000 

4,890 

In  knotty  wood 

Common  

In  clear  wood 

Screw 

In  knotty  wood 

Screw 

Art.  127. — Shearing  Resistance  of  Timber  behind  Bolt  or  Mortise 

Holes. 

Col.  T.  T.  S.  Laidley,  U.S.A.,  made  some  tests  during 
1 88 1  at  the  United  States  Arsenal,  Watertown,  Mass.,  on 
the  resistance  offered  by  timber  to  the  shearing  out  of  bolts 
or  keys  when  the  force  is  exerted  parallel  to  the  fibres. 


FIG.  i. 


FIG.  2. 


The  test  specimens  are  shown  in  Figs,  i  and  2.  Wrought- 
iron  bolts  and  square  wr ought-iron  keys  were  used.  All 
the  timber  specimens  were  six  inches  wide  and  two  inches 
thick.  The 'diameter  of  the  bolts  used  (Fig.  i)  was  one 
inch  for  all  the  specimens.  The  keys  were  i"Xi."5  and 
i". 1 25X1". 5  as  shown  in  Fig.  2.  In  all  the  latter  speci- 
mens, failure  took  place  in  front  of  the  smaller  key  where 
the  pressure  was  greatest. 


Art.  127.] 


SHEARING  RESISTANCE  OF  TIMBER. 


787 


In  many  cases  the  specimen  sheared  and  split  simultane- 
ously in  front  of  the  hole.  By  putting  bolts  through  the 
pieces  in  a  direction  normal  to  the  force  exerted,  so  as  to 
prevent  splitting,  the  resistance  was  found  (in  most  cases) 
to  be  considerably,  though  irregularly,  increased. 

Unless  otherwise  stated,  the  wood  was  thoroughly  sea- 
soned. 

The  accompanying  table  gives  the  results  of  Col.  Laid- 
lev's  tests. 


Kind  of  Wood. 

Centre 
of  Hole 
from 
End  of 
Speci- 
men. 

Total 
Area 

of 
Shearing. 

Ultimate  Shearing  Resistance 
per  Square  Inch,  in  Pounds. 

Ins. 

Sq.  Ins. 

r  2 

8 

399 

Spruce  (bolts) 

4 
'  6 

16 

24 

359 

275 

18 

32 

202 

f  2 

8 

457 

White  pine  (bolts). 

{* 

16 

24 

611 
450 

18 

32 

327 

f2 

8 

607 

Yellow  pine  (bolts)    

4 
6 

16 

24 

720 
456 

18 

32 

337- 

fa 

8 

599 

Yellow  pine  (square  keys)  . 

j  4 

!6 

16 

24 

369 

572 

17 

28 

438 

f2 

8 

550 

White  pine  (square  keys)  . 

u 

16 
24 

412 
332 

17 

28 

236 

2 

8 

410 

(Not  thoroughly  seasoned.) 

Spruce  (square  keys)  

.  4 

16 

24 

329 

242 

(Wet  timber.) 

17 

28 

279 

788  MISCELLANEOUS  SUBJECTS:  [Ch.  XVI. 

Art.  128.  —  Method  of  Least  Work  —  Stresses  in  a  Bridge  Portal. 

In  the  consideration  of  stresses  in  structures  or  parts  of 
structures  where  the  equations  of  condition  for  statical 
equilibrium  are  not  enough  to  determine  all  the  unknown 
quantities,  it  is  necessary  to  find  other  equations  involving 
the  elastic  properties  of  the  materials  used.  The  Method 
of  Least  Work  affords  one  procedure  by  which  such  extra 
equations  may  be  found. 

If  a  force  P  is  gradually  applied  at  a  point  in  a  struc- 
ture it  produces  a  deflection  or  distortion  5  in  its  own 
direction  and  performs  the  work, 


2a 


As  a  consequence  of  Hook's  law  P=ad,  a  being  a  con- 
stant and  a  direct  function  of  the  modulus  of  elasticity 
E  or  G.  Hence 

dW    P 


This  is  called  the  first  theorem  of  Castigliano,  enun- 
ciated in  his  "  Theorie  des  Gleichgewichtes  elastischer 
Systeme."  Eq.  (2)  is  perfectly  general  and  includes  all 
elastic  deformation  or  deflection.  It  shows  that  the  first 
derivative  of  W,  the  work  performed  by  the  load.  P,  in 
respect  to  that  load  as  the  independent  variable,  is  the 
elastic  distortion  as  well  in  the  case  of  a  force  acting  axially 
along  a  bar  either  in  tension  or  compression  as  in  that  of  a 
load  producing  deflection  of  a  bridge  at  its  point  of  applica- 
tion. 

The  third  member  of  eq.  (i)  shows  that 


Art.  128.]  STRESSES  IN  A  BRIDGE  PORTAL.  789 

This  equation  may  at  times  be  useful. 

If  the  point  of  application  of  the  force  or  load  P  in  eq. 
(2)  be  supposed  unchanged  in  position  while  P  acts,  the 
other  parts  of  the  structure  or  piece  moving  in  adjust- 
ment to  that  condition  as  may  be  required  by  the  corre- 
sponding strains,  then  will  5=o  and 

dW 


If  this  equation  be  satisfied  by  solving  it  for  P,  the 
resulting  value  will  make  W,  in  general,  either  a  maxi- 
mum or  minimum.  In  engineering  structures,  however, 
it  is  obvious  that  W  will  be  a  minimum,  as  the  test  by  the 
second  derivative  will  show  in  individual  cases. 

Eq.  (4)  expresses  Castigliano's  second  theorem.  If 
then  the  first  derivative  of  a  function  W  expressing  the 
work  performed  in  distorting  a  structure  or  structural 
member  in  terms  of  an  indeterminate  force  or  stress  P, 
whose  point  of  application  may  be  supposed  fixed,  be  taken 
in  reference  to  .  that  indeterminate  force  as  the  variable, 
a  new  equation  of  condition  will  result  whose  solution 
will  yield  a  value  of  the  force  making  the  energy  expended 
in  the  elastic  distortions  the  least  possible.  Hence  this 
procedure  is  called  "  the  method  of  least  work." 

Stresses  in  a  Bridge  Portal. 

The  treatment  of  a  bridge  portal  will  illustrate  the  use 
of  the  method  of  least  work  in  treating  an  important 
part  of  a  bridge.  Fig.  i  shows  a  skeleton  diagram  of  the 
portal,  AF  and  BG  being  the  end  posts  in  full  length  h  in 
their  own  plane.  ABCD  is  the  outline  of  the  portal  brac- 
ing which  may  be  a  plate  girder  or  open  bracing.  The 
corner  or  gusset  bracings  at  C  and  D  are  omitted.  The 


790 


MISCELLANEOUS  SUBJECTS. 


[Ch.  XVI. 


equal  end  post  stresses  due  to  vertical  dead  and  moving 
loads  are  indicated  by  P  and  P.  The  total  horizontal 
wind  load  acting  at  the  upper  ends  of  the  end  posts  is  shown 
by  H,  and  it  is  taken  as  applied  wholly  on  the  windward 
side.  As  is  usual,  the  end  posts  are  considered  fixed  in 
direction  at  both  upper  and  lower  ends.  The  lateral 


FIG.  i. 

action  of  the  wind  will  distort  the  portal  in  the  manner 
shown  exaggerated. 

As  both  posts  are  supposed  to  be  in  the  same  condition 
and  equally  affected  by  the  lateral  wind  pressure,  the  two 
points  of  contraflexure  K  and  0  must  be  at  same  distance 
ho  (to  be  determined)  from  FQ.  The  points  M'  and  M" 
are  in  the  neutral  surface  of  ABCD,  i.e.,  at  its  mid-depth. 
The  notation  of  Fig.  i  is  self  explanatory.  The  left  arrow 

TT 

—  is  below  K  and  external  to  the  upper  part  of  Fig.   i, 


Art.  128.1  STRESSES  IN  A  BRIDGE  PORTAL.  791 

TT 

but  the  right    arrow  —  is  above  0  and  external  to  the 

lower  part  of  the  figure.  Right-hand  moments  are  posi- 
tive and  left-hand  negative.  Taking  moments  of  forces 
acting  on  the  upper  part  ABOK  of  the  portal  and  about  0; 

(Pl-P=P')b-H(h-h0)=o     .'.     P'b=H(h-hQ).     (5) 

Obviously,.  P'  is  the  transferred  load  from  the  wind- 
ward truss  to  the  leeward  due  to  the  wind  pressure  H. 

In  order  to  find  the  work  performed  in  distorting  the 
members  of  the  portal,  it  is  necessary  to  determine  the 
bending  moments  M'  ',  M"  ',  Mi  and  Mi  at  the  points  indi- 
cated by  these  letters.  Taking  a  section  through  K  and 
moments  about  M',  Fig.  i  : 


...     M'=-(h-h0--}    .     .    ..  ...     .     (6) 

2  \  2/ 

Then  moments  about  M"  will  give 

M"  =  --(h-ho--}  =  -M'.  (7) 

2  \  2/ 

Obviously  the  signs  of  the  moments  Mi  and  Mi  must 
be  opposite  to  those  of  M"  and  M'  respectively.     Hence, 

M2=-/*o;    and    Mi  =  ~ho.     ...     (8) 

2  2 

The  moments  throughout  the  parts  of  the  portal  will 
then  be  : 


792  MISCELLANEOUS  SUBJECTS.  [Ch.  XVI. 

For  girder  AC, 


For  left  post  FA, 


For  right  post  BG, 

Mi-M' 
' 


---.     .     (n) 


•  It  has  been  shown  in  the  chapter  on  resilience  that  the 

work   done  in   bending   a  beam  is  —  ==  I  M2dx,   I  being 

z£rj 

the  moment  of  inertia  of  the  normal  section  of  the  beam. 
Similarly  the  work  performed  by  an  axial  force  P  on  a 
straight  member  whose  area  of  cross-section  is  A  and 

P2h 

length  h  is  .     If  /i  is  the   moment  of  inertia  for  the 

2/L.c, 

member  AC,  Fig.  i,  and  I2  for  each  post  AF  and  BG,  while 
A  2  is  the  common  area  of  cross-section  for  the  latter,  one 

77 

carrying    the    axial    load    P+  —  (h—ho)    and    the    other 

P 

77 

P—r(h—ho),  the  total  work  done  on  the  entire  portal  is 


2EI 


Art.  128.].  STRESSES  IN  A  BRIDGE  PORTAL.  793 

d  H2h2 

If  n=h—-  and  g  =P2-\  —  —  —  there  will  result: 
2  b2 


H2b  ,  f  H2k 

W  = 

3 


/n2 
[  -- 
\ 


h    I       2H2h7     ,  H2 

-- 


Eq.  (5)  shows  that  ho  may  be  replaced  in  this  equation 

f    D/       u  ^^  ^^        ni       • 

in  terms  of  P  ;    hence  -TT-  corresponds  to  -r^-..     Placing 


—  r—  =  o  and  solving,  therefore, 

O00' 


b3A2I2+6hb2A2I  '      ' 


This  locates  the  points  of  contraflexure  and  enables 
all  computations  to  be  made. 

If  the  axial  compression  of  the  two  end  posts  be  neglected, 
the  last  term  in  both  numerator  and  denominator  of  the 
second  member  of  eq.  (13)  disappears,  and 


l) 


If  T2  is  the  radius  of  gyration  of  A  2  and  if  -r  =i,  e 

-LI 

(13)  may  take  a  more  convenient  form  for  computation: 


794  MISCELLANEOUS  SUBJECTS.  [Ch.  XVI. 

In  the  same  manner  eq.  (14)  becomes: 


..      ....     (i6) 


l+6 


CHAPTER  XVII.  "'."/, 

•-  ;::  S 

THE  FATIGUE  OF  METALS. 

Art.  129.— Woehler's  Law. 

IN  all  the  preceding  pages,  that  force  or  stress  which, 
by  a  single  or  gradual  application,  will  cause  the  failure  or 
rupture  of  a  piece  of  material  has  been  called  its  "  ultimate 
resistance."  It  has  long  been  known,  however,  that  a  stress 
less  than  the  ultimate  resistance  may  cause  rupture  if  its 
application  be  repeated  (without  shock)  a  sufficient  number 
of  times.  Preceding  1859  no  experiments  had  been  made 
for  the  purpose  of  establishing  any  law  connecting  the  num- 
ber of  applications  with  the  stress  requisite  for  rupture,  or 
with  the  variation  between  the  greatest  and  least  values  of 
the  applied  stress. 

During  the  interval  between  1859  and  1870,  A.  Wohler, 
under  the  auspices  of  the  Prussian  Government,  undertook 
the  execution  of  some  experiments,  at  the  completion  of 
which  he  had  established  the  following  law: 

Rupture  may  be  caused  not  only  by  a  force  which  exceeds 
the  ultimate  resistance,  but  by  the  repeated  action  of  forces 
alternately  rising  and  falling  between  certain  limits,  the  greater 
of  which  is  less  than  the  ultimate  resistance;  the  number  of 
repetitions  requisite  for  rupture  being  an  inverse  function 
both  of  this  variation  of  the  applied  force  and  its  upper  limit. 

This  phenomenon  of  the  decrease  in  value  of  the  break- 

795 


796  THE  FATIGUE  OF  METALS.  fCh.  XVII. 

ing  load  with  an  increase  of  repetitions  is  known  as  "the 

fatigue  of  materials." 

Although  the  experimental  work  requisite  to  give 
Wohler 's  law  complete  quantitative  expression  in  the 
various  conditions  of  engineering  constructions  can  scarcely 
be  considered  more  than  begun,  yet  enough  has  been  done 
by  Wohler  and  Sparigenberg  to  establish  the  fact  of  metallic 
fatigue,  and  a  few  simple  formulae,  provisional  though  they 
may  be.  The  importance  of  the  subject  in  its  relation  to 
the  durability  of  all  iron  and  steel  structures  is  of  such  a 
high  character  that  a  synopsis  of  some  of  the  experimental 
results  of  Wohler  and  Spangenberg  will  be  given  in  the  next 
article. 

Art.  130. — Experimental  Resul  s. 

The  experiments  of  Wohler  are  given  in  "  Zeitschrift  fur 
Bauwesen,"  Vote.  X.,  XIII.,  XVI.,  and  XX.,  and  those  of 
Spangenberg  may  be  consulted  in  "Fatigue  of  Metals," 
translated  from  the  German  of  Prof.  Ludwig  Spangenberg, 
1876. 

These  results  show  in  a  very  marked  manner  the  effect 
of  repeated  vibrations  on  the  intensity  of  stress  required 
to  produce  rupture. 

Spangenberg  states  that  "the  experiments  show  that  vi- 
brations may  take  place  between  the  following  limits  with 
equal  security  against  rupture  by  tearing  or  crushing: 

r  +  i7,6oo  and— 17,600  Ibs.  per  sq.  in. 

Wrought  iron \  +  33,000  and —  o  "  "  ' 

I  +48,400  and +26,400  "  "  ' 

f +30,800  and— 30,800  "  "  " 

Axle  cast  steel 4  +  52,800  and  o  "  "  " 

I +88,000  and+ 38, 500  "  "  " 

f  +  55,ooo  and  o  "  "  " 

I  +77,000  and+  27, 500    "       "     " 
Spring-steel  not  hardened.  .  i    .  00  ,  , 

I  +88,000  and +  44,000    ' 

I +99,000  and +66; ooo    "       "     " 


Art.  130.)  EXPERIMENTAL   RESULTS. 

And  for  axle  cast  steel  in  shearing: 

+  24,200  and  — 24,200  Ibs.  per  sq.  in. 
+  41, 800  and    o   "   "  "  " 

PHCENIX  IRON  IN  TENSION. 


797 


Pounds  Stress  per 
Square  Inch. 

Number 
of  Repetitions. 

Pounds  Stress  per 
Square  Inch. 

Number 
of  Repetitions. 

o  to  52,800 
o  to  48,400 
o  to  44,000 
o  to  39,600 

800  rupture 
106,910  rupture 
340,853  rupture 
409,481  rupture 

o  to  39,600 
o  to  35,200 
22,000  to  48,400 
26,400  to  48,400 

480,852  rupture 
10,141,645  rupture 
2,373.424  rupture 
4,000,000  not  broken 

WESTPHALIA  IRON  IN  TENSION. 


o  to  52,800 

4,700  rupture 

o  to  39,600 

180,800  rupture 

o  to  48,400 

83,199  rupture 

o  to  39,600 

596)089  rupture 

o  to  48,400 

33,230  rupture 

o  to  39,600 

433,572  rupture 

o  to  44,000 

136,700  rupture 

o  to  35,200 

280,121  rupture 

o  to  44,000 

159,639  rupture 

o  to  35,200 

566,344  rupture 

FIRTH  &  SONS'  STEEL  IN  TENSION. 


o  to  66,000 

83,319  rupture     • 

o  to  55,000 

1  03  ,540  rupture 

o  to  60,500 

168,396  rupture 

o  to  53,900 

1  2,200,000  not  broken 

o  to  55,000 

133,910  rupture 

o  to  5  3,  goo 

229,230  rupture 

o  to  55,000 

185,680  rupture 

o  to  52,800 

692,543  rupture 

o  to  55,000 

360,235  rupture 

o  to  52,800 

12,200,000  not  broken 

o  to  55,000 

186,005  rupture 

o  to  50,600 

KRUPP'S  AXLE-STEEL  IN  TENSION. 


o  to  88,000 
o  to  77,000 
o  to  66,000 
o  to  60,500 

18,741  rupture 
46,286  rupture 
170,000  rupture 
1  23>77°  rupture 

o  to  55,000 
o  to  52,800 
o  to  50,600 

473,766  rupture 
13,600,000  not  broken 
1  2,200,000  not  broken 

PHOSPHOR-BRONZE  (UNWORKED)  IN  TENSIOT 


o  to  27,500 
o  to  22,000 
o  to  16,500 


147,850  rupture 

408,350  rupture 

2,731,161  rupture 


o  to  13,750 
o  to  13,750 


1,548,920  rupture 
2,340,000  rupture 


PHOSPHOR-BRONZE  (WROUGHT)  IN  TENSION. 


0  tO    22,000 

o  to  16,500 

53,900  rupture 
2,600,000  not  broken 

o  to  13,75° 

1,621,300  rupture 

798 


THE  FATIGUE  OF  METALS. 
COMMON  BRONZE  IN  TENSION. 


[Ch.  XVII. 


0  tO   22,000 

o  to  16,500 

4  200  rupture 
6,300  rupture 

o  to  11,000 

5,447,600  rupture 

PHCENIX  IRON  IN  FLEXURE  (ONE  DIRECTION  ONLY). 


Pounds  Stress  per 
Square  Inch. 

Number 
of  Repetitions. 

Pounds  Stress  per 
Square  Inch. 

Number 
of  Repetitions. 

o  to  60.500 
o  to  55,000 
o  to  49,500 
o  to  44,000 

169,750  rupture 
420,000  rupture 
481,975  rupture 
1,320.000  rupture 

o  to  39,600 
o  to  35,200 
o  to  33,000 

4,035,400  rupture 
3,420,000  rupture 
4,820,000  not  broken 

WESTPHALIA  IRON  IN  FLEXURE  (ONE  DIRECTION  ONLY). 


o  to  52,250 
o  to  49,500 
o  to  46.750 

612,065  rupture 
457,229  rupture 
799,543  rupture 

o  to  44,000 
o  to  39,600 

1,493,511  rupture 
3,587,509  rupture 

HOMOGENEOUS  IRON  IN  FLEXURE  (ONE  DIRECTION  ONLY). 


o  to  60,500 
o  to  55,000 
o  to  49,500 
o  to  44  ooo    - 

169,750  rupture 
420,000  rupture 
481,975  rupture 
1,320,000  rupture 

o  to  39,600 
°  to  35,020 
o  to  33,000 

4,035,400  rupture 
3,420,000  not  broken 
48,200,000  not  broken 

FIRTH  &  SONS'  STEEL  IN  FLEXURE  (ONE  DIRECTION  ONLY). 


o  to  63,250 
o  to  60,500 
o  to  55,000 

281,856  rupture 
266,556  rupture 
1,479,908  rupture 

o  to  52,250 
o  to  49,500 
o  to  49,500 

578,323  rupture 
5,640,596*  rupture 
1  3.  700,  ooo  not  broken 

*  Accidental. 


KRUPP'S  AXLE-STEEL  IN  FLEXURE  (ONE  DIRECTION  ONLY). 


o  to  77  ooo 
o  to  66.000 
c  to  60,500 


104,300  rupture 
317,275  rupture 
612,500  rupture 


o  to  55.000 
o  to  55,000 
o  to  49,500 


729,400  rupture 
1.499,600  rupture 
43,000,000  not  broken 


Art.  130.].  EXPERIMENTAL  RESULTS.  799 

KRUPP'S  SPRING-STEEL  IN  FLEXURE   (ONE  DIRECTION  ONLY). 


o  to  110,000 
o  to  88,000 
o  to  66,000 
o  to  55  ooo 
o  to  4Q.5oo 
88,000  to  132,000 
99,000  to  132,000 

39,950  rupture 
117,000  rupture 
468,200  rupture 
40,600,000  not  broken 
32,942,000  not  broken 
35,600,000  not  broken 
33.478,700  not  broker 

72,600  to  110,000 
66,000  to  99,000 
44,000  to  88,000 
44,000  to  88,000 
61,600  to  88,000 
27,500  to  77,000 
33,000  to  77,000 

19,673,300  not  broken 
33,600,000  not  broken 
35,800,000  not  broken 
38,000,000  not  broken 
36,000,000  not  broken 
36,600,000  not  broken 
31,152,000  not  broken 

PHOSPHOR-BRONZE  IN  FLEXURE  (ONE  DIRECTION  ONLY). 


Pounds  Stress  per 
Square  Inch. 

Number 
of  Repetitions. 

Pounds  Stress  per 
Square  Inch. 

Number 
of  Repetitions. 

o  to  22,000 
o  to  19,800 

862,980  rupture 
8,151,811  rupture 

o  to  16,500 
o  to  13,200 

5,075,169  rupture 
10,000,000  not  broken 

COMMON  BRONZE  IN  FLEXURE  (ONE  DIRECTION  ONLY). 


o  to  22,000 
o  to  19,800 

102,659  rupture 
151,310  rupture 

o  to  16,500 
o  to  13,200 

837,760  rupture 
10,400,000  not  broken 

PHCENIX  IRON  IN  TORSION  (BOTH  DIRECTIONS). 


—  35,200  to  +35,200 
—  43,000  to  +33  ooo 
—  28,600  to  +28  600 
—  26,400  to  +  26,400 

56,430  rupture 
99,000  rupture 
479,490  rupture 
909,810  rupture 

—  24,200  to  +24,200 

—  2  2,  000  tO    +22,000 

—  19,800  to  +  19.800 
—  17,600  to  +  17,600 

3,632,588  rupture 
4.917,992  rupture 
19  186,791  rupture 
132,250,000  not  broken 

ENGLISH  SPINDLE-IRON  IN  TORSION  (BOTH  DIRECTIONS). 


-37,400  to  +37,40- 
—  37,  400  to  +37,400 
-35.200  to  +35,200 
—  3  5,  200  to  +35,200 
-33,000  to  +33,000 
—  33  ooo  to  +  33.000 
—  30  800  to  +30,800 

204,400  rupture 
147,800  rupture 
9  1  1,  -i  oo  rupture 
402,900  rupture 
1,064,700  rupture 
384,800  rupture 
i»337,7oo  rupture 

—  30,  800  to  +  30,80- 
-28  600  to  +28,60 
—  28.600  to  +28,600 
—  26,400  to  +26,400 
—  26,  400  to  +26,400 

—  22,  000  tO    +22,000 
—  22  .000  tO    +  22,000 

979,100  rupture 
1,142,600  rupture 
595,910  rupture 
3,823,200  rupture 
6  100,000  not  broken 
8,800  ooo  not  broken 
4,000,000  not  broken 

KRUPP'S  AXLE-STEEL  IN  TORSION  (BOTH  DIRECTIONS) 


—  44,  ooo  to  +44,000 
—  39,600  to  +  39  6or 
—  37,400  to  +37,400 
—  35,  200  to  +35,200 
—  33,000  to  +  33  ooo 

367,400  rupture 
925,800  rupture 
4,900.000  not  broken 
4,800,000  not  broken 
5,000  ooo  not  broken 

-  46,200  to  +  4^.20 
—  37,  400  to  +  37,200 
—  35  200  to  +35,200 
—  33,000  to  +  33,000 
—  ?3,ooo  to  +33.000 

5  5,  roo  rupture 
797,525  rupture 
i  665,580  rupture 
4,163  375  rupture 
45,050.640  rupture 

I 

800  THE  FATIGUE  OF  METALS.  [Ch.  XVII. 

The  late  Capt.  Rodman,  U.SA.,  made  a  considerable 
number  of  experiments  on  the  fatigue  of  cast  iron,  but  they 
were  sufficient  in  number  and  character  to  show  the  general 
effect  only,  and  gave  no  quantitative  results. 

The  specimens  used  in  all  the  preceding  experiments 
were  small. 

During  1860,  '61,  and  '62  Sir  Wm.  Fairbairn  con- 
structed a  built  beam  of  plates  and  angles  with  a  depth  of 
1 6  inches,  clear  span  of  20  feet,  and  estimated  centre  break- 
ing load  of  26,880  pounds. 

This  beam  was  subjected  to  the  action  of  a  centre  load 
of  6643  pounds,  alternately  applied  and  relieved  eight 
times  per  minute;  596,790  continuous  applications  pro- 
duced no  visible  alterations. 

The  load  "was  then  increased  from  one  fourth  to  two 
sevenths  the  breaking  weight,  and  403,210  more  applications 
were  made  without  apparent  injury. 

The  load  was  next  increased  to  two  fifths  the  breaking 
weight,  or  to  10,486  pounds;  5175  changes  then  broke  the 
beam  in  the  tension  flange  near  the  centre. 

The  total  number  of  applications  was  thus  1,005,175. 

The  beam  was  then  repaired  and  loaded  with  10,500 
pounds  at  centre  158  times,  then  with  80*25  pounds  25,900 
times,  and  finally  with  6643  pounds  enough  times  to  make  a 
total  of  3,150,000. 

In  these  experiments  the  load  was  completely  removed 
each  time. 

It  is  thus  seen  that  vibrations  (without  shock)  with  one- 
fourth  the  calculated  breaking  centre  load  produced  no 
apparent  effect  on  the  resistance  of  the  beam,  but  that 
two  fifths  of  that  load  caused  failure  after  a  comparatively 
small  number  of  repetitions. 

It  is  probable  that  the  breaking  centre  load  was  calcu- 


Art.  131.]       FORMULAE  OF  LAUNHARDT  AND  H/EYRAUCH.  80 1 

lated  too  high,  in  which  case  the  ratios  J  and  f  should  be 
somewhat  increased. 

Art.  131. — Formulae  of  Launhardt  and  Weyrauch. 

Let  R  represent  the  intensity  (stress  per  square  unit  of 
section)  of  ultimate  resistance  for  any  material  in  tension, 
compression,  shearing,  torsion,  or  bending;  R  will  cause  rup- 
ture at  a  single,  gradual  application.  But  the  material  may 
also  be  ruptured  if  it  is  subjected  a  sufficient  number  of 
times,  and  alternately,  to  the  intensities  P  and  Q,  Q  being 
less  than  P  and  both  less  than  R,  while  all  are  of  the  same 
kind.  When  Q  =  o  let  P  =  W,  and  letD=P-Q.  W  is  called 
the  "primitive  safe  resistance,"  since  the  bar  returns  to  its 
primitive  unstressed  condition  at  each  application.  In  the 
general  case  P  is  called  the  "working  ultimate  resistance." 

By  the  notation  adopted: 

P='Q  +  D .    (i) 

But  by  Wohler's  law,  P  is  a  function  of  D,  or 

P=f(D).      ....     ;".     .     (2) 

A  sufficient  number  of  experiments  have  not  yet  been 
made  in  order  to  complete!  )r  determine  the  form  of  the 
function  /  (D). 

It  is  known,  however,  that 

•for.  Q=o,       P=D  =  W\ 
and  for  D=o,      P=  Q=R. 

Provisionally,  Launhardt  satisfies  these  two  extreme 
conditions  by  taking 


802  THE  FATIGUE  OF  METALS.  [Ch.  XVII. 


Even  at  these  limits  this  is  not  thoroughly  satisfactory, 

when  D=o,  P  =  °( 
o 

By  solving  eq.  (3), 


for  when  D=o,  P  =  °(R-W),  or  is  indeterminate. 
o 


But  if  the  least  value  of  the  total  stress  to  which  any 
member  of  a  structure  is  subjected  is  represented  by  min  B, 

and  its  greatest  value  by  max  B,  there  will  result  -  75  =  p. 
Hence 

R-WminB 


which  is  Launhardt's  formula.  In  the  preceding  article 
some  values  of  W  are  shown.  In  applying  eq.  (5)  it  is  only 
necessary  to  take  the  primitive  safe  resistance,  W,  for  the 
total  number  of  times  which  the  structure  will  be  subjected 
to  loads.  Since  bridges  are  expected  to  possess  an  indefinite 
duration  of  life,  in  such  structures  that  number  should  be 
indefinitely  large, 

Eq.  (5),  it  is  to  be  borne  in  mind,  is  to  be  applied  when 
the  piece  is  always  subjected  to  stress  of  one  kind,  or  in  one 
direction  only.  It  agrees  well  with  some  experiments  by 
Wohler  on  Krupp's  untempered  cast  spring  steel. 

If  the  stress  in  any  piece  varies  from  one  kind  to  another, 
as  from  tension  to  compression,  or  vice  versa,  or  from  one 
direction  to  another,  as  in  torsion  on  each  side  of  a  state  of 
no  stress,  Weyrauch  has  established  the  following  formula 
by  a  course  of  reasoning  similar  to  that  used  by  Launhardt. 

If  the  opposite  stresses,  which  will  cause  rupture  by  a 
certain  number  of  applications,  are  equal  in  intensity,  and 


Art.  131.]       FORMULA  OF  LAUNHARDT  AND  WEYRAUCH.  803 

if  that  intensity  is  represented  by  5,  then  will  5  be  called 
the  "  vibration  resistance"  ;  this  was  established  by  Wohler 
for  some  cases,  and  some  of  its  values  are  given  in  the  pre- 
ceding article. 

Let  +  P  and  —  Pf  represent  two  intensities  of  opposite 
kinds  or  in  opposite  directions,  of  which  P  is  numerically  the 
greater.  Then  if  D  =  P  +  P'  , 

P=D-P'. 

The  two  following  limiting  conditions  will  hold: 

For  P'=o,     P=D  =  W\ 
ForP'=S;    P  =  S  =  *2D. 

But  by  Wohler'  s  law  P=f(D),  and  the  two  limiting 
conditions  just  given  will  be  found  to  be  satisfied  by  the 
provisional  formula 

w,s  W_S 

-2W-S-PU~2\V-S-P^ 

By  the  solution  of  eq.  (6), 


If,  without  regard  to  kind  or  direction,  max  B  is  numer- 
ically the  greatest  total  stress  which  the  piece  has  to  carry, 
while  max  Bf  is  the  greatest  total  stress  of  the  other  kind 

Pf     max  Bf 

or  direction,  then  will  -75-  =  -     ~.     Hence  there  will  result 
P      max  B 

the  following,  which  is  the  formula  of  Weyrauch: 


W      max 


804  THE  FATIGUE  OF  METALS.  [Ch.  XVII. 

Eqs.  (5)  and  (8)  give  values  of  the  intensity  P  which  are 
to  be  used  in  determining  the  cross-section  of  pieces  d  e- 
signed  to  carry  given  amounts  of  stress.  If  n  is  the  safety 
factor  and  F  the  total  stress  to  be  carried,  the  area  of  sec- 
tion desired  will  be 

A_nF_ 
A  ~:  p  » 

p  t 
in  which  —  is  the  greatest  working  stress  permitted. 

If  for  wrought  iron  in  tension  W  =  30,000  and  R  = 
50,000,  eq.  (5)  gives 


/       2  mm  B  \ 
P  =  30,000   i  +-      — 7-,). 
\       3  max  B] 


Hence,  if  the  total  stress  due  to  fixed  and  moving  loads 
in  the  web  member  of  a  truss  is  max  B  =  80,000  pounds, 
while  that  due  to  the  fixed  load  alone  is  min  B=  40,000, 
there  will  result 


/       2 
(1  +.- 


=30,0001     .-.  =40,000. 


In  such  a  case  the  greatest  permissible  working  stress 
with  a  safety  factor  of  3  would  be  about  13,300  pounds. 
For  steel  in  tension,  if  W  =  50,000  and  ^  =  75,000, 

T-,  /i  min  B  \ 

P  =  50,000   i  +-        -=). 
\       2  max  B/ 

For  wrought  iron  in  torsion,  if  5  =  18,000  and  W  =  24,000, 
eq.    (8)  will  give 


/       i  max  Bf\ 
\       4  max  B  / 


Art.  132.]  INFLUENCE  OF  TIME  ON  STRAINS.  805 

Other  methods  based  on  Wohler's  experiments  have  been 
deduced  by  Muller,  Gerber,  and  Schaffer,  of  which  synopses 
may  be  found  in  Du  Bois'  translation  of  Weyrauch's 
"  Structures  of  Iron  and  Steel." 


Art.  132. — Influence  of  Time  on  Strains. 

In  an  earlier  section  of  this  book  devoted  to  data  of 
certain  tests,  the  effect  of  prolonged  tensile  stress  and 
subsequent  rest  between  the  elastic  limit  and  ultimate  resist- 
ance was  shown  to  be  the  elevation  of  both  those  quantities. 
It  is  a  matter  of  common  observation,  however,  that  if  a  piece 
of  wrought  iron  be  subjected  to  a  tensile  stress  nearly  equal 
to  its  ultimate  resistance,  and  held  in  that  condition,  the 
stretch  will  increase  as  the  time  elapses. 

Experiments  are  still  lacking  which  may  show  that  a 
piece  of  metal  can  be  ruptured  by  a  tensile  stress  much 
below  its  ultimate  resistance.  It  may  be  indirectly  inferred, 
however,  from  experiments  on  flexure,  that  such  failure 
may  be  produced,  as  the  following  by  Prof.  Thurston  will 
show. 

A  bar  10  parts  tin  and  90  parts  copper,  1X1X22  inches 
and  supported  at  each  end,  sustained  about  65  per  cent,  of 
its  breaking  load  at  the  centre  for  five  minutes.  During 
that  time  its  deflection  increased  0.021  inch.  The  same 
bar  sustained  1485  pounds  at  centre  for  13  minutes  and 
then  failed. 

A  second  bar  of  the  same  size,  but  90  parts  tin  and  10 
parts  copper,  was  loaded  at  the  centre  with  160  pounds, 
causing  a  deflection  of  1.294  inches.  After  10  minutes  the 
deflection  had  increased  0.025  inch ;  after  one  day,  i  .00  inch ; 
after  two  days,  2.00  inches ;  and  after  three  days,  3.00  inches, 
when  the  bar.  failed  under  the  load  of  160  pounds. 

Another  bar  of  the  same  size  showed  remarkable  results ; 


806  THE  FATIGUE  OF  METALS.  [Cn.  XVII 

it  was  composed  of  90  parts  zinc  and  10  parts  copper.  It 
gave  the  same  general  increase  of  deflection  with  time,  but 
eventually  broke  under  a  centre  load  which  ran  down  from 
1233  to  911  pounds,  after  holding  the  latter  about  three 
minutes. 

A  bar  of  the  same  size  and  96  parts  copper  with  4  parts 
tin,  after  it  had  carried  700  pounds  at  centre  for  sixty  min- 
utes was  loaded  with  1000  pounds,  with  the  following 
results : 

After.  Deflection. 

o  minute 3.118  inches, 

5  minutes 3 . 540 

1 5  minutes 3 . 660  " 

45  minutes 4 . 102  " 

75  minutes 7 . 634  " 

Broke  under  1000  pounds. 

A  wrought-iron  bar  of  the  same  size  gave,  under  a  centre 
load  of  1600  pounds: 

After.  Deflection. 

o  minute o .  489  inch. 

3  minutes o .  632 

6  minutes o .  650 

16  minutes o .  660 

344  minutes o .  660 

It  subsequently  carried  2589  pounds  with  a  deflection  of 
4.67  inches. 

During  1875  and  1876  Prof.  Thurston  made  a  number  of 
other  similar  experiments  with  the  same  general  results. 

Metals  like  tin  and  many  of  its  alloys  showed  an  increas- 
ing rate  of  deflection  and  final  failure,  far  below  the  so-called 
"ultimate  resistance."  The  wrought-iron  bars,  however, 
showed  a  decreasing  increment  of  deflection,  which  finally 
became  zero,  leaving  the  deflection  constant. 

Whether  there  may  be  a  point  for  every  metal,  beyond 


Art.  132.]  INFLUENCE  OF  TIME  ON  STRAINS.  807 

which,  with  a  given  load,  the  increment  of  deflection  may 
retain  its  value  or  go  on  increasing  until  failure,  and  below 
which  this  increment  decreases  as  the  time  elapses,  and 
finally  becomes  zero,  is  yet  undetermined,  but  seems  proba- 
ble. 

It  does  not  follow,  therefore,  that  the  principle  enunci- 
ated in  the  section  named  at  the  beginning  of  this  article 
is  to  be  taken  without  qualification.  If  "rest"  under 
stress,  too  near  the  ultimate  resistance,  be  sufficiently  pro- 
longed, it  has  been  seen  that  it  is  possible  that  failure  may 
take  place. 

In  verifying  some  experimental  results  by  Herman 
Haupt,  determined  over  forty  years  ago,.  Prof.  Thurston 
tested  three  seasoned  pine  beams  about  ij  inches  square 
and  40  inches  length  of  span,  and  found  that  60  per  cent, 
of  the  ordinary  "breaking  load"  caused  failure  at  the  end 
of  8,  12,  and  15  months.  In  these  cases  the  deflection  slowly 
and  steadily  increased  during  the  periods  named. 

Two  other  sets  of  three  pine  beams  each  broke  under  80 
and  95  per  cent,  of  the  usual  "breaking  load,"  after  much 
shorter  intervals  of  time. 

In  all  these  instances  it  is  evident  that  the  molecules 
under  the  greatest  stress  "  flow"  over  each  other  to  a  greater 
or  less  extent.  In  the  cases  of  decreasing  increments  of 
strain,  the  new  positions  afford  capacity  of  increased  resist- 
ance ;  in  the  others,  those  movements  are  so  great  that  the 
distances  between  some  of  the  molecules  exceed  the  reach 
of  molecular  action,  and  failure  follows. 

In  many  cases  strained  portions  of  material  recover  par- 
tially or  wholly  from  permanent  set.  In  such  cases  a  por- 
tion of  the  material  has  been  subjected  to  intensities  of 
stress  high  enough  to  produce  true  "  flow"  of  the  molecules, 
while  the  remaining  portion  has  not.  The  internal  elastic 
stresses  in  the  latter  portion,  after  the  removal  of  the  exter- 


8o8  THE  FATIGUE  OF  METALS.  [Ch.  XVII. 

nal  forces,  produce  in  time  a  reverse  flow  in  consequence  of 
the.  elastic  endeavor  to  resume  the  original  shape. 

It  is  altogether  probable  that  the  phenomena  of  fatigue 
and  flow  of  metals  are  very  intimately  associated.  Some 
of  the  prominent  characteristics  of  the  latter  will  be  given 
in  the  next  chapter 


CHAPTER  XVIII. 

THE  FLOW  OF  SOLIDS. 

Art.  133. — General  Statements. 

ALTHOUGH  there  is  no  reason  to  suppose  that  true  solids 
may  not  retain  a  definite  shape  for  an  indefinite  length  of 
time  if  subjected  to  no  external  force  other  than  gravity,* 
many  phenomena  resulting  both  from  direct  experiment  for 
the  purpose,  and  incidentally  from  other  experiments  involv- 
ing the  application  of  external  stress  of  considerable  inten- 
sity, show  that  a  proper  intensity  of  internal  stress  (in 
many  cases  comparatively  low)  will  cause  the  molecules  of  a 
solid  to  flow  at  ordinary  temperatures  like  those  of  a  liquid. 
And  this  flow,  moreover,  is  entirely  different  from,  and  inde- 
pendent of,  the  elastic  properties  of  the  material;  for  it 
arises  from  a  permanent  and  considerable  relative  displace- 
ment of  the  molecules.  Nor  is  it  to  be  confounded  with 
that  internal  ''friction"  which,  if  an  elastic  body  is  sub- 
jected to  oscillations,  causes  the  amplitudes  to  gradually 
decrease  and  finally  disappear,  even  in  vacuo.  This  latter 
motion  is  typically  elastic  and  the  retarding  cause  may  be 
considered  a  kind  of  elastic  friction. 

It  is  evident  that  if  a  mass  of  material  be  enclosed  on  all 
its  faces,  or  outer  surfaces,  but  one  or  a  portion  of  one,  and 
if  external  pressure  be  brought  to  bear  on  those  faces,  the 

*This,  perhaps,  may  be  considered  a  definition  of  a  true  solid. 

809 


8io 


THE  FLOW  OF  SOLIDS. 


[Ch.  XVIII. 


p:x~E 

G  
B  — 

_______ 

snzii 

D            H 

O 

K 

C 

FIG.  i. 


r 


b 


material  will  be  forced  to  move  to  and  through  the  free  sur- 
face; in  other  words,  the  flow  of  the  material  will  take  place 
in  the  direction  of  least  resistance. 

The  theory  of  the  flow  of  solids 
to  be  given  is  that  developed  by 
Mons.^H.  Tresca  in  his  "  Memoire 
sur  1'Ecoulement  des  Corps  So- 
lides,"  1865.  He  made  a  large 
number  of  experiments  on  hard 
and  soft  metals,  ceramic  pastes, 
sand,  and  shot. 

These  different  materials  all 
manifested  the  same  characteris- 
tics of  flow,  which  are  well  shown 
in  Fig.  2.  ABCD,  Fig.  i,  is  sup- 
posed to  be  a  cylindrical  mass  of 
lead  with  circular  horizontal  sec- 
tion, confined  in  a  circular  cylin- 
der, MA7,  closed  at  one  end  with 
the  exception  of  the  orifice  0. 

This  cylinder  is  supported  on 
the  base  PA',  while  the  face  AB 
of  the  lead  receives  external  pres- 
sure from  a  close-fitting  piston. 
When  the  pressure  is  sufficiently 
increased,  the  face  AB  in  Fig.  i 
sinks  to  A  B  in  Fig.  2,  while  the 
column  hkHK,  in  the  latter  figure, 
is  forced  to  flow  through  the  ori- 
fice  0. 

In  Tresca' s  experiments  with 

lead,  the  diameter  A  B  was  about  3.9  inches;  the  diameter 
HK  of  the  orifice,  from  0.75  in.  to  1.5  ins.,  while  the  length 
of  the  column  or  jet  hK  varied  from  0.4  in.  to  about  24  ins. 


Art,  134.]  TRESCA'S  HYPOTHESES.  8ll 

The  total  pressure  on  the  face  AB  varied  from  119,000  to 
198,000  pounds.  The  initial  thickness  AD  varied  from  0.24 
inch  to  2.4  inches. 

Some  experiments  exhibiting  in  a  remarkably  clear  man- 
ner the  flow  of  metals  in  cold  punching  were  made  by  David 
Townsend  in  1878,  and  the  results  were  given  by  him  in  the 
"  Journal  of  the  Franklin  Institute"  for  March  of  that  year. 
If  the  dotted  rectangle  ABFG,  Fig.  3,  shows  the  original 
outline  of  the  middle  section  of  a  nut  before  punching,  he 
found  that  the  final  outline  of  the  same  section  would  be 
represented  by  the  full  lines.  The  top  and  bottom  faces 
were  depressed  by  the  punching,  as  shown ;  the  upper  width 
A  B  remained  about  the  same,  but  the  lower,  GF,  was  in- 
creased to  CD.  Although  the  depth  of  the  nut,  AC,  was  1.75 
inches,  the  length  of  the  core  punched  out  was  only  1.063 
inches.  The  density  of  this  core  was  then  examined  and 
found  to  be  the  same  as  that  of  the  original  nut.  Hence  a 
portion  of  the  core  equal  in  length  to  1.75  —  1.063=0.687 
inch  was  forced,  or  flowed,  back  into  the  body  of  the  nut. 
Subsequent  experiments  showed  that  this  flow  did  not  take 
place  at  the  immediate  upper  surface  AB,  nor  very  much 
in  the  lower  half  of  the  nut,  but  that  it  was  chiefly  confined 
to  a  zone  equal  in  depth  to  about  half  that  of  the  nut,  the 
upper  surface  of  which  lies  a  very  short  distance  below  the 
upper  face  of  the  nut.  The  location  of  this  zone  is  shown  by 
the  lines  HK  and  MA7  in  Fig.  3. 

Tresca's  experiments  on  punching  showed  essentially  the 
same  result. 

Art.  134. — Tresca's  Hypotheses. 

The  central  cylinder  FGKH>  Fig.  i  of  Art.  133  was  called 
by  Tresca  the  "primitive  central  cylinder."  As  the  metal 
flows,  this  cylinder  will  be  drawn  out  into  the  volume  of 
revolution,  whose  axis  is  that  of  the  orifice  and  whose 


8.*2  THE  FLOW  OF  SOLIDS.  [Ch.  XVIII. 

meridian  section  is  FGkKHh,  Fig.  2,  the  diameter  FG  being 
gradually  decreased. 

It  was  found  by  experiment  that  if  the  original  mass  AC, 
Fig.  i ,  was  composed  of  horizontal  layers  of  uniform  thick- 
ness, the  reduced  mass  in  Fig.  2  was  also  composed  of  the 
same  number  of  layers  of  uniform  thickness,  except  in  the 
immediate  vicinity  of  the  central  cylinder. 

Tresca  then  assumed  these  three  hypotheses: 
i°. — The  density  of  the  material  remains  the  same  whether 
in  the  cylinder  or  in  the  jet;  in  other  words,  the  volume  of  the 
material  in  the  jet  and  in  the  cylinder  remains  constant. 

Let  R  =  radius  of  the  cylinder; 
Rl  =  radius  of  the  orifice ; 
y  =yariable  length  of  the  jet  (i.e.,  hH)\ 
D  =  original  depth  of  material  (BC  =AD,  Fig.  i) 

in  the  cylinder; 

d  =  variable  depth  of  material  (BC  =  AD,  Fig.  2) 
in  the  cylinder; 

then  by  the  hypothesis  just  stated 

R^d=R2D-R^y .     (i) 

2°. — The  rate  of  compression  along  any  and  all  lines  paral- 
lel to  the  axis  of  the  primitive  central  cylinder,  and  taken  outside 
of  that  limit,  is  constant. 

If,  then,  the  material  lying  outside  of  the  central  cylinder 
be  divided  into  horizontal  layers  of  equal  thickness,  a  very 
small  decrease  in  the  variable  depth  equal  to  d  (a)  will  cause 
the  same  amount  of  material  to  move  or  now  from  each  of 
these  layers  into  the  space  originally  occupied  by  the  central 
cylinder,  thus  causing  a  portion  of  the  material  previously 
resting  over  the  orifice  to  flow  through  the  latter.  If  d(d) 
i-j  the  indefinitely  small  change  of  depth,  and  dR1  the  in- 
definitely small  change  in  the  radius  of  the  cylindrical  por- 


Art.  135.]  THE   VARIABLE  MERIDIAN  SECTION.  813 

tion  resting  over  the  orifice,  then  the  equality  of  volumes 
expressing  this  hypothesis  is  the  following: 


or 

d(d)      2RidRl 


(2) 


3°. — The  rate  of  decrease  of  the  radius  of  the  primitive  cen- 
tral cylinder  is  constant  throughout  its  length  at  any  given  in- 
stant during  flow. 

Let  r  be  any  radius  less  than  Rv  then  if  the  latter  is  de- 
creased by  the  very  small  amount  dRly  the  former  will  be 
shortened  by  the  amount  dr\  and  by  the  last  hypothesis 
there  must  result 


dR.     dr 


R 


(3) 


This  is  a  perfectly  general  equation,  in  which  r  may  or 
may  not  be  the  variable  value  of  the  radius  of  that  portion 
of  the  primitive  central  cylinder  remaining  above  the  orifice 
at  any  instant  during  now. 

These  are  the  three  hypotheses  on  which  Tresca  based 
his  theory  of  the  flow  of  solids.  It  is  thus  seen  to  be  put 
upon,  a  purely  geometrical  basis,  entirely  independent  of  the 
elastic  or  other  properties  of  the  material. 

+ 

Art.    135.— The  Variable    Meridian    Section    of    the    Primitive 
Central  Cylinder. 

The  meridian  curve  haH,  or  hbK,  Fig.  2  of  Art.  133, 
may  now  easily  be  determined. 

Eq.  (i)  of  Art.  134  may  take  the  first  of  the  following 


814  THE  FLOW  OF  SOLIDS.  [Ch.  XVIII. 

forms,  while  its  differential,  considering  d  and  y  variable, 
may  take  the  second: 


Dividing  the  second  by  the  first, 
d(d)          dy          2R1 


The  last  member  of  this  equation  is  simply  eq.  (2)  of 
Art.  134;  and  if  the  value  of  dRv  in  eq.  (3)  of  the  same 
article,  be  inserted,  in  the  third  member  of  this  equation, 
there  will  result 

27?^      dr          dy 


R 


Integrating  between  the  limits  of  r  and  Rv  and  remem- 
bering that  r  will  be  restricted  to  the  representation  of  the 
radius  of  that  portion  of  the  primitive  central  cylinder 
which  remains,  at  any  instant,  over  the  orifice,  by  taking 
y  =  o  for  r  =  Rv 


K*      .       r      .      / '     K. 

1         in&  —  — I  no  I    • 

r>  2  l°B    ZP    —WS  02 


R 


"log"  indicates  a  Napierian  logarithm. 


Art.  136.]     POSITIONS  IN  THE  JET  Of-  HORIZONTAL  SECTIONS.   815 

Passing  from  logarithms  to  the  quantities  themselves, 
and  reducing, 


^  n 

« 


This  is  the  desired  equation  of  the  line,  in  which  r  is 
measured  normal  to  the  axis  of  the  cylinder  or  jet,  while  y 
is  measured  along  that  axis  from  the  extremity  of  the  jet. 
When  the  material  is  wholly  expelled, 

R2 


Eq.  (2)  is  applicable  to  the  jet  only.    For  the  line  hF  or 
Gk,  resort  will  be  had  to  the  equation 

d(d)  =     2RS     dr 
d    ~R*-R^  r' 

Again  integrating  between  the  limits  d  and  D,  or  r  and 
Rv  and  reducing, 


This  value  of  r  is  the  radius  of  that  portion  of  the  primi- 
tive central  cylinder  which  remains  over  the  orifice  when  D 
is  reduced  to  d. 

Art.  136. — Positions  in  the  Jet  of  Horizontal  Sections  of  the 
Primitive  Central  Cylinder. 

That  portion  of  the  primitive  central  cylinder  below  ab, 
in  Fig.  i  of  Art.  133  will  be  changed  to  abKH  in  Fig.  2  of 
the  same  article. 


8i6  THE  FLOW  OF  SOLIDS.  [Ch.  XVIII. 

If,  in  the  latter  Fig.,  y'  is  the  distance  from  HK  to  ab, 
measured  along  the  axis,  then  the  volume  of  HKab  will 
have  the  value 

/y 
r.r^dy. 
o 

If  df  is  the  distance  aF  =  bG,  in  Fig.  i,  the  equality  of 
volumes  will  give 

I**  r*dy  =  R*(D-dr). 
J   o 

Eq.  (i)  of  Art.  125  gives 


/ 
r*dy=R*D  -  R 


If  N  is  the  number  of  horizontal  layers  required  to  com- 
pose the  total  thickness  D,  and.  n  the  number  in  the  depth  d'  , 


D     A/- 
Hence 


C*-l 
/      \   R2 
-W  > 


Art.  137.]         FINAL  RADIUS  OF  HORIZONTAL  SECTION.  817 

Tresca  computed  values  of  y'  for  some  of  his  experiments 
and  compared  the  results  with  actual  measurements.  The 
agreement,  though  not  exact,  was  very  satisfactory.  Within 
limits  not  extreme,  the  longer  the  jet  the  more  satisfactory 
was  the  agreement. 


Art.  137. — Final  Radius  of  a  Horizontal  Section  of  the  Primitive 
Central  Cylinder. 

Let  it  be  required  to  determine  what  radius  the  section 
situated  at  the  distance  df  from  the  upper  surface  of  the 
primitive  central  cylinder  will  possess  in  the  jet. 

It  will  only  be  necessary  to  put  for  y  in  eq.  (i)  of  Art. 
135  the  value  of  yr  taken  from  eq..  (i)  of  Art.  136.  This 
operation  gives 


Hence 

/rjf\      2R* 

(i) 


If  7\1  is  small,  as  compared  with  R,  there  will  result  ap- 
proximately 


Art.  138.  —  Path  of  Any  Molecule. 

The  hypotheses  on  which  the  theory  of  flow  is  based 
enable  the  hypothetical  path  of  any  molecule  to  be  easily 
established. 


8i8  THE  FLOW  OF  SOLIDS.  [Ch.  XVIII. 

In  consequence  of  the  nature  of  the  motion  there  will  be 
three  portions  of  the  path,  each  of  which  will  be  represented 
by  its  characteristic  equation,  as  follows: 

First,  let  the  molecule  lie  outside  of  the  primitive  central 
cylinder. 

Let  Rf  and  H  be  the  original  co-ordinates  of  the  mole^ 
cule  considered,  measured  normal  to  and  along  the  axis  of 
the  cylinder,  respectively,  from  the  centre  of  the  orifice  HK 
(Fig.  i,  Art.  133)  as  an  origin,  while  r  and  h  are  the  variable 
co-ordinates. 

The  first  hypothesis,  by  which  the  density  remains  con- 
stant, then  gives  the  following  equation: 


or 

This  is  the  equation  to  the  path  of  the  molecule,  in 
which  r  must  always  exceed  Rr 

As  this  equation  is  of  the  third  degree,  the  curve  cannot 
be  one  of  the  conic  sections. 

Second,  let  the  molecule  move  in  the  space  originally  occu- 
pied by  the  central  cylinder. 

While  h  and  r  now  vary,  the  volume  nr2(D  —  h)  must 
remain  constant.  When  r  =  Rl  let  h  =-hr  Hence 

r2(D-h)=R12(D-h1) (2) 

But  if  h=h1  and  r  =  R^  in  eq.  (i), 


Placing  this  value  in  eq.  (2). 

...     (3) 


Art.  138.]  PATH  OF  ANY  MOLECULE.  8iq 

Third,  let  the  molecule  move  in  the  jet. 

After  the  molecule  passes  the  orifice,  its  path  will  evi- 
dently be  a  straight  line  parallel  to  the  axis  of  the  jet.  Its 
distance  rt  from  that  axis  will  be  found  by  putting  h  =  o  in 
eq.  (3).  Hence 

H  R2-R'Z\* 


APPENDIX  I. 

ELEMENTS  OF  THEORY  OF  ELASTICITY  IN 
AMORPHOUS  SOLID  BODIES, 


CHAPTER  I. 
GENERAL   EQUATIONS. 

Art.  i. — Expressions  for  Tangential  and  Direct  Stresses  in  Terms 
of  the  Rates  of  Strains  at  Any  Point  of  a  Homogeneous  Body. 

LET  any  portion  of  material  perfectly  homogeneous  be 
subjected  to  any  state  of  stress  whatever.  At  any  point  as 
0,  Fig.  i,  let  there  be  assumed  any  three  rectangular  co- 
ordinate planes;  then  consider  any  small  rectangular  par- 
allelepiped whose  faces  are  parallel  to  those  planes.  Finally 
let  the  stresses  on  the  three  faces  nearest  the  origin  be  re- 
solved into  components  normal  and  parallel  to  their  planes 
of  action,  whose  directions  are  parallel  to  the  co-ordinate 
axis. 

The  intensities  of  these  tangential  and  normal  compo- 
nents will  be  represented  in  the  usual  manner,  i.e.,  /^signi- 
fies a  tangential  intensity  on  a  plane  normal  to  the  axis  of 
X  (plane  ZY),  whose  direction  is  parallel  to  the  axis  of 
y,  while  pxx  signifies  the  intensity  of  a  normal  stress  on 

820 


Art.  i.] 


TANGENTIAL  AND  DIRECT  STRESSES. 


821 


a  plane  normal  to  the  axis  of  X  (plane  ZY)  and  in  the 
direction  of  the  axis  of  X.  Two  unlike  subscripts,  there- 
fore, indicate  a  tangential  stress,  while  two  of  the  same  kind 
signify  a  normal  stress. 


FIG.  i. 

From  eq.   (3),  Art.  2,  and  eq.   (7),  Art.  5,  there  is  at 
once  deduced 


«$  = 


2(1 


(i) 


Now  when  the  material  is  subjected  to  stress  the  lines 
bounding  the  faces  of  the  parallelepiped  will  no  longer  be 
at  right  angles  to  each  other.  It  has  already  been  shown 
in  Art.  2  that  the  angular  changes  of  the  lines  from  right 
angles  are  the  characteristic  shearing  strains,  which,  multi- 
plied by  G$  give  the  shearing  intensities. 

Let  (^  be  the  change  of  angle  of  the  boundary  lines 
parallel  to  X  and  Y. 

Let  <j)2  be  the  change  of  angle  of  the  boundary  lines 
parallel  to  Y  and  Z. 

Let  </>3,  be  the  change  of  angle  of  the  boundary  line 
parallel  to  Z  and  X. 


822  ELASTICITY  IN  AMORPHOUS  SOLID  BODIES.  [Ch.  I. 

Eq.  (i)  will  then  give  the  following  three,  equations: 


M 


In  Fig.  i  let  the  rectangle  agfh  represent  the  right  pro- 
jection of  the  indefinitely  small  parallelepiped  doc  dy  dz.  If 
u,  v,  and  w  are  the  unit  strains  parallel  to  the  axes  of  x,  y, 
and  z  of  the  original  point  h,  the  rates  of  variation  of  strain 

du    dv    dw  -  ., 

-r-,  -7-,  -  —  ,  etc.,  may  be  considered  constant  throughout 

dx    dy    dz 

this  parallelepiped;  consequently  the  rectangular  faces  will 
change  to  oblique  parallelograms.  The  oblique  parallelo- 
gram dhck,  whose  diagonals  may  or  may  not  coincide  with 
those  of  agfh,  therefore,  may  represent  the  strained  con- 
dition of  the  latter  figure. 

Then,  by  Art.  2,  the  difference  petween  dhc  and  the  right 
angle  at  h  will  represent  the  strain  0r  But,  from  Fig.  i,  <^ 
has  the  following  value: 


(5) 


But  the  limiting  values  of  the  angles  in  the  second  mem- 
ber are  coincident  with  their  tangents  ;  hence 

de      be 


Art.  i.]  STRESSES  IN  TERMS  OF  STRAINS.  823 

But,  again,  de  is  the  distortion  parallel  to  OX  found  by 
moving  parallel  to  OY  only;  hence  it  is  a  partial  differential 
of  u,  or  it  has  the  value 


In  precisely  the  same  manner  be  is  the  partial  differential 
of  v  in  respect  to  x,  or 

L      dv  j 
bc  =  -r-dx. 
dx  • 

By  the  aid  of  these  considerations,  eq.  (6)  takes  the  form 

du     dv 


If  XY  be  changed  to  YZ,  and  then  to  ZXy  there  may  be 
at  once  written  by  the  aid  of  eq.  (8) 

dv     dw 

^=dz+Ty>  -  ;  •  ;   •   •     (9) 

dw    du 


Eqs.  (2),  (3),  and  (4)  now  take  the  following  form: 

dv 


d 


824  ELASTICITY  IN   AMORRHOUS  SOLID  BODIES.        [Ch.  I. 

The  direct  stresses  are  next  to  be  given  in  terms  of  the 
displacements  u,  v,  and  w.  Again,  let  the  rectangular  par- 
allelepiped dx  dy  dz  be  considered.  Eq.  (i),  on  page  3, 
shows  that  the  strain  per  unit  of  length  is  found  by  dividing 
the  intensity  of  stress  by  the  coefficient  of  elasticity,  if  a  sin- 
gle stress  only  exists.  But  in  the  present  instance,  any  state 
of  stress  whatever  is  supposed.  Consequently  the  strain 
caused  by  pxxt  for  example,  acting  alone  must  be  combined 
with  the  lateral  strains  induced  by  pyy  and  pgg.  Denoting 
the  actual  rates  of  strain  along  the  axes  of  X,  Y,  and  Z  by 
lv  12,  and  /3,  therefore,  the  following  equations  may  be  at  once 
written  by  the  aid  of  the  principles  given  on  pages  9  and  10  : 


ds) 


Eliminating  between  these  three  equations, 


But  if  w,  v,  and  w  are  the  actual  strains  at  the  point  where 
these  stresses  exist,  the  rates  of  strain  lv  lv  and  13  will  evi- 


Art.   i.]  STRESSES  IN   TERMS  OF  STRAINS.  825 

du  dv         dw 
dently  be  equal  to  -T-,  -T-  ,  and  -77,  respectively.     The  volume 

of  the  parallelepiped  will  be  changed  by  those  strains  to 
dx(i  -\-l^dy(\  +I2)dz(i  +/3)  =dx  dy  dz(i  +  /1  +  /2  +  /3) 

if  powers  of  lv  /2,  and  /3  above  the  first  be  omitted.  The 
quantity  (/1  +  /2  +  y  is,  then,  the  rate  of  variation  of  volume, 
or  the  amount  of  variation  of  volume  for  a  cubic  unit.  If 
there  be  put 

du     dv     dw  E 

o=-r-  +-r-  +-?—,     and     (j=—, 


eqs.  (17),  (18),  and  (19)  wrill  take  the  forms 

^du 

+  2GV;       .     .     .     .'    (20) 
y 


i  —  2r 


(22) 


The  form  in  which  eqs.  (14),  (15),  and  (16)  are  written 
shows  that  if  pxv,  pyy,  or  pzs  is  positive,  the  stress  is  tension, 
and  compression  if  it  is  negative.  Consequently  a  positive 
value  for  any  of  the  intensities  in  eqs.  (20),  (21),  or  (22)  will 
indicate  a  tensile  stress,  while  a  negative  value  will  show 
the  stress  to  be  compressive. 

The  eqs.  (14)  to  (19),  together  with  the  elimination  in- 
volved, also  show  that  the  coefficients  of  elasticity  for  ten- 
sion and  compression  have  been  taken  equal  to  each  other, 
and  that  the  ratio  r  is  the  same  for  tensile  and  compressive 
strains. 


826  ELASTICITY  IN  AMORPHOUS  SOLID  BODIES.         [Ch.  I. 

Further,  in  eqs.  (n),  (12),  and  (13),  it  has  been  assumed 
that  G  is  the  same  for  all  planes. 

Hence  eqs.  (n,)  (12),  (13),  (20),  (21),  and  (22)  apply 
only  to  bodies  perfectly  homogeneous  in  all  directions. 

It  is  to  be  observed  that  the  co-ordinate  axes  have  been 
taken  perfectly  arbitrarily. 

Art.  2.  —  General  Equations  of  Internal  Motion  and  Equilibrium. 

In  establishing  the  general  equations  of  motion  and  equi- 
librium, the  principles  of  dynamics  and  statics  are  to  be 
applied  to  the  forces  which  act  upon  the  parallelepiped  repre- 
sented in  Fig.  i  ,  the  edges  of  which  are  doc,  dy,  and  dz.  The 
notation  to  be  used  for  the  intensities  of  the  stresses  acting 
on  the  different  faces  will  be  the  same  as  that  used  in  the 
preceding  article. 

Let  the  stresses  which  act  on  the  faces  nearest  the  origin 
be  considered  negative,  while  those  which  act  on  the  other 
three  faces  are  taken  as  positive. 

The  stresses  which  act  in  the  direction  of  the  axis  of  X 
are  the  following: 

On  the  face  normal  to  X,  nearest  to       0,  —  pxjt  dy  dz  ; 

"     "      "         "       "    "  farthest  from  0,  (pxx  +  -^dxj 

\  ax      / 


"     "      " 


dy  doc  nearest  to  0,  -pzx  dy  dx\ 


"     "      "     "    "    farthest  from  0,  ( psx  +  -%**dz}dydx\ 


"     dz  doo  nearest  to  0,  —  pyx  dz  dx\ 

"      "    "    farthest  from          0. 


dz 


^ 

.'•''az 

dy 
dx 

dx 

,-'"' 

^.^ 

dy 


Art.  2.]     EQUATIONS  IN  RECTANGULAR  CO-ORDINATES.  827 

The  differential  coefficients  of  the  intensities  are  the  rates 
of  variation  of  those  intensities  for  each  unit  of  the  variable, 
which,  multiplied  by  the 
differentials  of  the  varia- 
bles, give  the  amounts  of 
variation  for  the  different 
edges  of  the  parallelepiped. 

Let  X0  be  the  external 
force  acting  in  the  direc- 
tion of  X  on  a  unit  of  vol- 
ume at  the  point  consid- 
ered ;  then  X0dxdy  dz  will 
be  the  amount  of  external 
force  acting  on  the  paral- 
lelopiped. 

These  constitute  all  the  forces  acting  on  the  parallelo- 
piped  in  the  direction  of  the  axis  of  X,  and  their  sum,  if  un- 

•  d2u 
balanced,  must  be  equal  to  m-r^dx  dy  dz ;  in  which  m  is  the 

mass  or  inertia  of  a  unit  of  volume,  and  dt  the  differential 
of  the  time.  Forming  such  an  equation,  therefore,  and  drop- 
ping the  common  factor  doc  dy  dz,  there  will  result 


FIG.  i. 


, 

~ 


=m^.      .     .     .     (i) 


Changing  x  to  y,  y  to  z,  and  z  to  x,  eq.  (i)  will  become 


-W-—  (l} 

—  mj*2'  •         •         •          \2) 


dx        dy        dz 
Again,  in  eq.  (i),  changing  x  to  z,  z  to  y,  and  y  to  x, 
dp       dp  yz     dp. 


dx       dy        dz 


T5-        -     -     -      (3) 


828  ELASTICITY  IN  AMORPHOUS  SOLID  BODIES.         [Ch.  I. 

The  line  of  action  of  the  resultant  of  all  the  forces  which 
act  on  the  indefinitely  small  parallelepiped,  at  its  limit, 
passes  through  its  centre  of  gravity,  consequently  it  is  sub- 
jected to  the  action  of  no  unbalanced  moment.  The  parallele- 
piped, therefore,  can  have  no  rotation  about  an  axis  passing 
through  its  centre  of  gravity,  whether  it  be  in  motion  or 
equilibrium.  Hence,  let  an  axis  passing  through  its  centre 
of  gravity  and  parallel  to  the  axis  of  X,  be  considered.  The 
only  stresses,  which,  from  their  direction  can  possibly  have 
moments  about  that  axis,  are  those  with  the  subscripts  (yz), 
(zy),  (yy),  or  (zz).  But  those  with  the  last  two  subscripts 
act  directly  through  the  centre  of  the  parallelepiped,  conse- 

quently their  moments  are  zero.    The  stresses  -^r*dy  •  doc  dz 

and     jZy  dz  .  dx  dy  are  two  of  six  forces  whose  resultant  is 

directly  opposed  to  the  resultant  of  those  three  forces  which 
represent  the  increase  of  the  intensities  of  the  normal,  or 
direct,  stresses  on  three  of  the  faces  of  the  parallelepiped; 
these,  therefore,  have  no  moments  about  the  assumed  axis. 
The  only  stresses  remaining  are  those  whose  intensities  are 
pzy  and  pyz.  The  resultant  moment,  which  must  be  equal 
to  zero,  then,  has  the  following  value: 

xdy.dz  =  o'1       ...     (4) 

P2y  .......        (5) 


Hence  the  two  intensities  are  equal  to  each  other. 

The  negative  sign  in  eq.  (5)  simply  indicates  that  their 
moments  have  opposite  signs  or  directions;  consequently, 
that  the  shears  themselves,  on  adjacent  faces,  act  toward 
or  from  the  edge  between  those  faces.  In  eqs.  (i),  (2),  and 
(3),  the  tangential  stresses,  or  shears,  are  all  to  be  affected 


Art.  2.]     EQUATIONS  IN  RECTANGULAR  CO-ORDINATES.  829 

by  the  same  sign,  since  direct,  or  normal,  stresses  only  can 
have  different  signs. 

The    eq.  (5)  is  perfectly  general,  hence  there  may  be 
written  : 

Pxy=Pyx>  and  pzx=pxz.     .  •••.  "V   ;     (6) 
Adopting  the  notation  of  Lame,  there  may  be  written: 


by  which  eqs.  (i),  (2),  and  (3)  take  the  following  forms: 
dNl     dT3    dT 


m-df'>     •    •    •    (7) 

dT3     dN2    dT, 

+  V  +  ~^  +  F°  = 

dT2     dT,     dN3 

- 


The  equations  (u),  (12),  (13),  (20),  (21),  and  (22)  of  the 
preceding  article  are  really  kinematical  in  nature  ;  in  order 
that  the  principles  of  dynamics  may  hold,  they  must  satisfy 
eqs.  (7),  (8),  and  (9).  As  the  latter  stand,  by  themselves, 
they  are  applicable  to  rigid  bodies  as  well  as  elastic  ones; 
but  when  the  values  of  N  and  T,  in  terms  of  the  strains  u,  v, 
and  w,  have  been  inserted,  they  are  restricted,  in  their  use, 
to  elastic  bodies  only.  With  those  values  so  inserted,  they 
form  the  equations  on  which  are  based  the  mathematical 
theory  of  sound  and  light  vibrations,  as  well  as  those  of 
elastic  rods,  membranes,  etc.  In  general,  they  are  the  equa- 
tions of  motion  which  the  different  parts  of  the  body  can 


830  ELASTICITY  IN  AMORPHOUS  SOLID  BODIES.         [Ch.  I. 

have  in  reference  to  each  other,  in  consequence  of  the  elastic 
nature  of  the  material  of  which  the  body  is  composed. 

If  all  parts  of  the  body  are  in  equilibrium  under  the 
action  of  the  internal  stresses,  the  rates  of  variation  of  the 


d2u    d2v         , 
strains  -ITF,  -r^,  and  -^-,  will   each   be    equal  to  zero. 

Hence,  eqs.  (7),  (8),  and  (9)  will  take  the  forms 

dN.     dTs     dT, 

o;.     •     •     •     do) 


dTa     dN,    dT. 


.       ..... 

dx       dy       dz 

These  are  the  general  equations  of  equilibrium.  As  they 
stand,  they  apply  to  a  rigid  body.  For  an  elastic  body,  the 
values  of  N  and  T  from  the  preceding  article,  in  terms  of  the 
strains  u,  v,  and  w,  must  satisfy  these  equations. 

The  eqs.  (10),  (n),  and  (12)  express  the  three  conditions 
of  equilibrium  that  the  sums  of  the  forces  acting  on  the 
small  parallelepiped,  taken  in  three  rectangular  co-ordinate 
directions,  must  each  be  equal  to  zero.  The  other  three  con- 
ditions, indicating  that  the  three  component  moments  about 
the  same  co-ordinate  axes  must  each  be  equal  to  zero,  are 
fulfilled  by  eqs.  (5)  and  (6).  The  latter  conditions  really 
eliminate  three  of  the  nine  unknown  stresses.  The  remaining 
six  consequently  appear  in  both  the  equations  of  motion 
and  equilibrium. 

The  equations  (7)  to  (12),  inclusive,  belong  to  the  interior 
of  the  body.  At  the  exterior  surface,  only  a  portion  of  the 
small  parallelepiped  will  exist,  and  that  portion  will  be  a 


Art.  2.]      EQUATIONS  IN  RECTANGULAR  CO-ORDINATES.  831 

tetrahedron,  the  base  of  which  forms  a  part  of  the  exterior 
surface  of  the  body,  and  is  acted  upon  by  external  forcea 

Let  —  be  the  area  of  the  base  of  this  tetrahedron,  and  let 

p,  q,  and  r  be  the  angles  which  a  normal  to  it  forms  with 
the  three  axes  of  X,  Y,  Z,  respectively.  Then  will 

da  cos  p  =  dy  dz,  da  cos  q=dz  dx,  and  da  cos  r  =  dx  dy. 

Let  P  be  the  known  intensity  of  the  external  force  acting 
on  da,  and  let  TT,  /,  and  p  be  the  angles  which  its  direction 
makes  with  the  co-ordinate  axes.  Then  there  will  result  : 

X0  =  P  da  ..  cos  TT,  Y0=PJa.cos/,  and  Z0=Pda.cosp. 

The  origin  is  now  supposed  to  be  so  taken  that  the  apex  of 
the  tetrahedron  is  located  between  it  and  the  base;  hence 
that  part  of  the  parallelepiped  in  which  acted  the  stresses 
involving  the  derivatives,  or  differential  coefficients,  is 
wanting  ;  consequently  those  stresses  are  also  wanting. 

The  sums  of  the  forces,  then,  which  act  on  the  tetra- 
hedron, in  the  co-ordinate  directions,  are  the  following: 

-  (N\  dy  dz  +  T3  dz  dx  +  T2  dy  dx)  +  Pda  cos  TT  =  o; 

—  (T3  dz  dy  +  A/3  dz  dx  +  7\  dy  dx)  +  Pda  cos  7  =  0; 

—  (T2  dz  dy  +  7\  dz  dx  +  N3  dy  dx)  +  Pda  cos  p  =  o. 

Substituting  from  above, 

Nl  cos  p  +  T3  cos  q  +  T2  cos  r  =  P  cos  TT  ;  .  .  (13) 

T3  cos  p  +  N2  cos  q  +  Tl  cos  r  =  P  cos  #  ;  .  .  (14) 

T2  cos  p  +  Tl  cos  q  +  N3  cos  r  **  P  cos  p.  ,  4  (15) 


These  equations  must  always  be  satisfied  at  the  exterior 
surface  of  the  body;  and  since  the  external  forces  must 
always  be  known,  in  order  that  a  problem  may  be  determi- 
nate, they  will  serve  to  determine  constants  which  arise 


832 


ELASTICITY  IN  AMORPHOUS  SOLID  BODIES. 


[Ch.  I. 


from  the  integration  of  the  general  equations  of  motion  and 
equilibrium. 

Art.  3. — Equations  of  Motion  and    Equilibrium  in  Semi-polar 

Co-ordinates. 

For  many  purposes  it  is  convenient  to  have  the  condi- 
tions of  motion  and  equilibrium  expressed  in  either  semi- 
polar  or  polar  co-ordinates ;  the  first  form  of  such  expression 
will  be  given  in  this  article. 

The  general  analytical  method  of  transformation  of  co- 
ordinates may  be  applied  to  the  equations  of  the  preceding 
article,  but  the  direct  treatment  of  an  indefinitely  small 
portion  of  the  material,  limited  by  co-ordinate  surfaces,  pos- 
sesses many  advantages.  In  Fig.  i  are  shown  both  the 


FIG.  i. 

small  portion  of  material  and  the  co-ordinates,  semi-polar 
as  well  as  rectangular.  The  angle  made  by  a  plane  normal 
to  ZY,  and  containing  OX,  with  the  plane  XY  is  repre- 
sented by  <j> ;  the  distance  of  any  point  from  OX,  measured 
parallel  to  ZY,  is  called  r\  the  third  co-ordinate,  normal  to 


Art.  3.]         EQUATIONS  IN  SEMI-POLAR  CO-ORDINATES.  833 

r  and  <£,  is  the  co-ordinate  xt  as  before.  It  is  important  to 
observe  that  the  co-ordinates  x,  r,  and  <£,  at  any  point,  are 
rectangular. 

The  indefinitely  small  portion  of  material  to  be  con- 
sidered will,  as  shown  in  Fig.  i ,  be  limited  by  the  edges  dx,  dr, 
and  r  d<j>.  The  faces  dx  dr  are  inclined  to  each  other  at  the 
angle  d(j>. 

The  intensities  of  the  normal  stresses  in  the  directions  of 
X  and  r  will  be  indicated  by  A/\  and  R,  respectively.  The 
remainder  of  the  notation  will  be  of  the  same  general  char- 
acter as  that  in  the  preceding  article;  i.e.,  Txr  will  represent 
a  shear  on  the  face  dr.rd(j)m  the  direction  of  r,  while  N^  is 
a  normal  stress,  in  the  direction  of  <£,  on  the  face  dx  dr. 

The  strains  or  displacements,  in  the  directions  of  x,  r,  and 
(f>,  will  be  represented  by  u,  p,  and  w\  consequently  the 
unbalanced  forces  in  those  directions,  per  unit  of  mass, 
will  be 

dzu         d2p  d*w  ,  x 

mdT"    mW'  and   mW  •    '    •    •    (l) 

Those  forces  acting  on  the  faces  hf,  fe,  and  he,  will  be 
considered  negative ;  those  acting  on  the  other  faces,  posi- 
tive. 

Forces  Acting  in  ike  Direction  of  r. 

—  R.rd<f>dx,  and 

+  Rrd<j>dx+  (~|p<fr  =  rjjdr +Rdr\(l<j>  dx. 

—  T^dr  dx,  and 

+  T^dr  dx  +  ^rd<l> .  dr  dx. 

—  Txr .  r  dcj)  dr,  and 


834  ELASTICITY  IN  AMORPHOUS  SOLID   BODIES.         [Ch.  I. 

On  the  face  dr  dx,  nearest  to  ZOX,  there  acts  the  normal 

stress  (N^drdx-\---r^d^.drdx}=Nr\    and  N'  has  a  com- 
\  ^r  / 

ponent  acting  parallel  to  the  face  fe  and  toward  OX,  equal  to 
N'  sin  (d<j>)  =  N'T—^=N'd(l>.  But  the  second  term  of  this 

product  will  hold  (d(f>)2,  hence  it  will  disappear,  at  the  limit, 
in  the  first  derivative  of  N'd<j>  /.  Nrd^  =  N^d(j)  dr  doc. 
Since  this  force  must  be  taken  as  acting  toward  OX,  it 
acts  with  the  normal  forces  on  hf,  and,  consequently,  must 
be  given  the  negative  sign. 

If  RQ  is  the  external  force  acting  on  a  unit  of  volume, 
another  force  (external)  acting  along  r  will  be  RQ  .  r  d<t>  dr  dx. 

The  sum  of  all  these  forces  will  be  equal  to 

m.  rd<j>  dr  dx  .  ~. 


Forces  Acting  in  ike  Direction  of  $. 
dx,  and 

+  N^dr  dx  +  d-^d$  .  dr  djc. 
—  Trj  .  r  d(f>  dx,  and 


dr,  and 
/7T" 


As  in  the  case  of  NW,  in  connection  with  the  forces  along 
r,  so  the  force  T^  dr  dx  has  a  component  along  </>  (normal 
to  fe)  equal  to  T^drdx.  sin  (d$)  =T^rd^  dr  dx.  It  will 
have  a  positive  sign,  because  it  acts  from  OX. 

The  external  force  is  @0.r  d<j>  dr  dx. 


Art.  3-]         EQUATIONS  IN  SEMI-POLAR  CO-ORDINATES.  835 

Forces  Acting  in  the  Direction  of  x. 

-ATj.r  d(j>  dr,  and 

dN 
f  NjT  d(j>dr  +  -j-^dx  .  r  d<j>  dr. 

~Trx.dx  r  d$,  and 


dr,  and 
+  T6xdx  dr  +  ~d^>  .  doc  dr. 


Th?  external  force  is  X0  .  r  d<j>  dx  dr. 

Putting  each  of  these  three  sums  equal  to  the  proper 
rates  of  variation  of  momentum,  and  dropping  the  common 
factor  r  d<j>  dx  dr: 


These  are  the  general  equations  of  motion  (vibration)  in 
terms  of  semi-polar  co-ordinates  ;  if  the  second  members  are 
made  equal  to  zero,  they  become  equations  of  equilibrium. 
Eqs.  (2),  (3),  and  (4),  are  not  dependent  upon  the  nature  of 
the  body. 

Since  x,  r,  and  <j>  are  rectangular,  it  at  once  follows  that 

Trx  =  Txr,  Tr<f>  =  Tfr,  and  Tx+  =  T^.       .     .     (5) 


836  ELASTICITY  IN  AMORPHOUS  SOLID  BODIES.        [Ch.  I. 

In  order  that  eqs.  (2),  (3),  and  (4)  may  be  restricted  to 
elastic  bodies,  it  is  necessary  to  express  the  six  intensities 
of  stresses  involved,  in  terms  of  the  rates  of  variation  of  the 
strains  in  the  rectangular  co-ordinate  directions  of  x,  r,  and 
(j>.  Since  these  co-ordinates  are  rectangular,  the  eqs.  (n), 
(12),  (13),  (20),  (21),  and  (22)  of  Article  i,  may  be  made 
applicable  to  the  present  case  by  some  very  simple  changes 
dependent  upon  the  nature  of  semi-polar  co-ordinates. 

For  the  present  purpose  the  strains  in  the  co-ordinate 
directions  of  x,  y,  and  z  will  be  represented  by  u',  v'  ',  and 
wf.  Since  the  axis  of  x  remains  the  same  in  the  two  systems, 
evidently 

dur  _du 

doc  ~  doc* 

From  Fig.  i  it  is  clear  that  the  axis  of  y  corresponds 
exactly  to  the  co-ordinate  direction  r\  hence 


= 
dy     dr' 

From  the  same  Fig.  it  is  seen  that  the  axis  of  z  corre- 
sponds to  <£,  or  T(f>.  But  the  total  differential,  dwf  ,  must  be 
considered  as  made  up  of  two  parts  ;  consequently  the  rate 

of  variation  -j-  will  consist  of  two  parts  also.    If  there  is  no 

distortion  in  the  direction  of  r,  or  if  the  distance  of  a  mole- 
cule from  the  origin  remains  the  same,  one  part  will  be 

-77  —  —  =—  -fi.    If>  however,  a  unit's  length  of  material  be  re- 
d(r$)     rd<f> 

moved  from  the  distance  r  to  r+  p  from  the  centre  0,  Fig.  i, 
while  <j>  remains  constant,  its  length  will  be  changed  from 

i   to    fi+—  j,   in  which    p  may  be  implicitly  positive   or 


Art.  3.  ]        EQUATIONS  IN  SEMI-POLAR  CO-ORDINATES.  837 

negative.     Consequently  there  will  result 

dwf  _  dw      p 
dz  ~rd<j>     r' 

For  the  reason  already  given,  there  follow 

du'  du  .  dvf  dp 
-T^=~5 —  and  -3 — '  =  ~r~. 
dy  dr  dx  doc 

In  Fig.  2  let  dc  be  the  side  of  a  distorted  small  portion 
of  the  material,  the  original  position  Q  af  e 

of  which  was  d'e.  Od  is  the  distance 
r  from  the  origin,  ad=dr  and  ac  = 
dw,  while  ddf  =w.  The  angular 

ac     dw 
change  in  position  of  dc  is  ~~(r*~(r\  FIG.  2. 

but  an  amount  equal  to  —3  =  —  is  due  to  the  movement  of 

r,  and  is  not  a  movement  of  dc  relatively  to  the  material 
immediately  adjacent  to  d. 
Hence 

dwf  _dw     w  dvf      dp 

lty~dr~~r>     also     dz~=7d$' 

There  only  remain  the  following  two,  which  may  be  at 
once  written 

dw'     dw  duf      du 

—r-  =  T—     and     — r- 


dx     dx  dz     rd<f>' 

The  rate  of  variation  of  volume  takes  the  following  form 
in  terms  of  the  new  co-ordinates: 

(W     dv      du/du    dp      dw      p 


838         ELASTICITY  IN  AMORPHOUS  SOLID  BODIES.  LCh-  *• 

Accenting  the  intensities  which  belong  to  the  rectan- 
gular system  x,  y,  z,  the  eqs.  (n),  (12),  (13),  (20),  (21),  and 
).  of  Art.  i,  take  the  following  form: 


».  -»;-+><     •••••« 


dw 


If  these  values  are  introduced  in  eqs.  (2),  (3),  and  (4), 
those  equations  will  be  restricted  in  application  to  bodies 
of  homogeneous  elasticity  only. 

The  notation  t  is  used  to  indicate  that  the  r  involved  is 
the  ratio  of  lateral  to  direct  strain,  and  that  it  has  no  rela- 
tion whatever  to  the  co-ordinate  r. 

The  limiting  equations  of  condition,  (13),  (14),  and  (15) 
of  Art.  2,  remain  the  same,  except  for  the  changes  of  nota- 
tion, shown  in  eqs.  (7)  to  (12),  for  the  intensities  N  and  T. 


Art.  4.] 


EQUATIONS  IN  POLAR  CO-ORDINATES. 


839 


Art.  4. — Equations  of  Motion  and  Equilibrium  in  Polar 
Co-ordinates. 

The  relation,  in  space,  existing  between  the  polar  and 
rectangular  systems  of  co-ordinates  is  shown  in  Fig.  i .  The 
angle  </>  is  measured  in  the  plane  ZY  and  from  that  of  XY ; 


FIG.  i. 

while  </»  is  measured  normal  to  ZY  in  a  plane  which  contains 
OX.  The  analytical  relation  existing  between  the  two  sys- 
tems is,  then,  the  following: 


oc  =  r  sin  </>,     y  =  r  cos  <p  cos 


and 


=  r  cos     sn 


The  indefinitely  small  portion  of  material  to  be  considered 
is  a  h  e  d.  It  is  limited  by  the  co-ordinate  planes  located  by 
</>  and  0,  and  concentric  spherical  surfaces  with  radii  r  and 
r  +  dr.  The  directions  r,  <£,  and  0,  at  any  point,  are  rectangu- 
lar ;  hence  the  sums  of  the  forces  acting  on  the  small  portion 
of  the  material,  taken  in  these  directions,  must  be  found  and 
put  equal  to 

msf'  mdt>'  and  mW' 


840  ELASTICITY  IN  AMORPHOUS  SOLID  BODIES.          [Ch.  I. 

in  which  expressions,  p,  t),  and  w  represent  the  strains  in  the 
direction  of  r,  </>,  and  ^  respectively. 

Those  forces  which  act  on  the  faces  ah,  bd,  and  cd  will  be 
considered  negative,  and  those  which  act  on  the  other  faces 
positive. 

The  notation  will  remain  the  same  as  in  the  preceding 
articles,  except  that  the  three  normal  stresses  will  be  indi- 
cated by  Nry  Nf,  and  A/V 

Forces  Acting  Along  r. 
—  Nr.r  dfy  r  cos  $  d<f>. 
-fATr.r2cos  <[>  d$  d<j> 

+  2rNdr   cos 


rdr\ 


dT^ 

—  T^.rcos  $  d(j)  dr. 
.r  cos  <{r  d(f>  dr 

ft. 


j, 

—  N$  .r  d<p  dr.  sin  aOc  —  —  N^>  .  r  d  </t  dr  .  cos  <p  d(f>,  on  face  ce. 

—  AT^.rcos  <[>  d$  dr.smaOb  =  —Nf.r  cos  <[>  d$  dr.d<l>, 

on  face  be. 

Forces  Acting  Along  $. 

—  Tr<3?  .  r  cos  </>  d<j>  r  d  0. 


cos  0  rf#  d<t>. 


Art.  4.J  EQUATIONS  IN  POLAR  CO-ORDINATES. 

—  N<J>  .  r  d  (f>  dr. 


—  T^.r  cos  0  d(j>  dr. 
+  7^cos  <f>.r  d(j>  dr 

/J(T^cos  <[>)  .  .  tdTt+j  r     T*      •      t  j  \    j  i  j 

+  (-     ^         —  -d<l>=cos  (ff-r^d^-T^  sin  ^  d<f>  jr  d<f>  dr. 

+  T$rrd  </;  dr  .  cos  0  d<£,  on  face  c^. 


m  akc  =  '  =  -Trddr.  sin 


on  face  ce. 
The  lines  a^  and  c&  are  drawn  normal  to  Oc  and  Oa. 

Forces  Acting  Along  <p. 
—  Tr.r  cos  <f>  d<j>.r  di[>. 


d(Trd>r'1}  , 

H       a    =      —T  r<l, 


,  rj  „     , 

ar  =      —T  dr+2rTr<l,dr}cos 


\ 

} 


-T^.rd^dr. 


—  NJ,  .  r  cos  $  d(j)  dr. 
.r  cos 


d        ps  N  d 


\-Tfr.r  cos  <f>d<t>dr.d</>,  on  face  6^. 

f  A^0  .rd(I>dr.smakc=*  +  N^  r  d</>dr.sm  $  d<j>,  on  face 


842  ELASTICITY  IN  AMORPHOUS  SOLID  BODIES.          [Ch.  I. 

The  volume  of  the  indefinitely  vSmall  portion  of  the 
material  is  (omitting  second  powers  of  indefinitely  small 
quantities) 

r  cos  <l>  d(f> .  r  d <[> . dr  =  J  V, 

and  its  mass  is  m  multiplied  by  this  small  volume.  The 
latter  may  be  made  a  common  factor  in  each  of  the  three 
sums  to  be  taken. 

The  external  forces  acting  in  the  directions  R,  <£,  and  ^ 
will  be  represented  by 


and 
respectively. 

Taking  each  of  the  three  sums,  already  mentioned,  and 
dropping  the  common  factor  JV,  there  will  result 

dNr         dT6r       .  AT*,  ,  2A>-.V(i-A^-r^tan 
~jr~~r 


dr      r  cos  (l>.d(j> 

=  W^72">         C1) 


V        I  V  T  TT 

dr       r  cos  ^.^^     rc/^ 

+ *± 1 ^  ^r 1_^LJ±±Lr  +  0        w^v.      ^^ 


+  - VV7  + 


Jr      r  cos  0<i^     r  d0 

2  T^  +  7^  -  N#  tan  0  +  N^  tan  ^ 


^  y^,  (a) 


^ 

Since  r,  •£,  and  <[>  are  rectangular  at  any  point, 
7    =  7,     and     T^T. 


Art.  4.)  EQUATIONS  IN  POLAR  CO-ORDINATES.  843 

Hence 


These  relations  somewhat  simplify  the  first  members  df 
eqs.  (2)  and  (3). 

Eqs.  (i),  (2),  and  (3)  are  entirely  independent  of  the 
nature  of  the  material  ;  also,  they  apply  to  the  case  of  equi- 
librium, if  the  second  members  are  made  equal  to  zero. 

The  rectangular  rates  of  strain,  at  any  point,  in  terms 
of  r,  (j>,  and  ^  are  next  to  be  found.  As  in  the  preceding 
article,  the  rates  of  strain  in  the  rectangular  directions  of 
r,  </>,  and  ^  will  be  indicated  by 

dv'    dwf   duf   dv'   duf 
dy"  ~dzT'  dx?'dxnjy"  etc* 

Remembering  the  reasoning  in  connection  with  the  value 
of  ~j—  ,  in  the  preceding  article,  and  attentively  considering 
Fig.  i,  there  may  at  once  be  written, 

du'  _  da)      p 


In  Fig.  i,  if  ac  =  i  and  ab  =  CD,  while  ak=r  cot.  $  (oik  is 
perpendicular  to  aO),  the  difference  in  length  between  ac 
and  bh  will  be 

w  a>  tan  ^ 

r  cot  <l>  ~          r 

This  expression  is  negative  because  a  decrease  in  length  takes 
place  in  consequence  of  a  movement  in  the  positive  direction 
of  r<f>. 


844 


ELASTICITY  IN  AMORPHOUS  SOLID  BODIES.         [Ch.  I. 


Again,  a  consideration  of  Fig.  i,  and  the  reasoning  con- 
nected with  the  equation  above,  will  give 

dw'  dt]  p     a)tan(p 


p 
r 


Without  explanation  there  may  at  once  be  written  : 

djS     dp 

dyf~dr' 

Fig.  i  of  this,  and  Fig.  2  of  the  /preceding  article,  give 


duf     d<D     aj  dvf 

=~ 


dp 


These  are  to  be  used  in  the  expression  for  T^. 
the  same  figures  and  method  give 


Precisely 


dv' 


dp 


r  cos  $  d<j> 


dw      dy     y 
and     -j-f  =-3 — -, 
dy      dr     r1 


which  are  to  be  used  in  finding  T(f>r. 

dw' 
,  The  expression  for  -7-7-  will  be  composed  of  the  sum  of 

two  parts.  In  Fig.  2,  ab  is  the  original  position  of  r  d<{>,  and 
after  the  strain  y  exists  it  takes  the  position  ec.  Consequently 
*  ac  (equal  and  parallel  to  bd  and  perpen- 

dicular to  ak)  represents  the  strain  r), 
while  ed  represents  dy.  Since,  also,  fc  is 
perpendicular  to  ck,  the  strains  of  the  kind 
y  change  the  right  angle  fck  to  the  angle 
fce-t  or  the  angle  eck  is  equal  to 

dw'  ed     ca 

j-r  =  ecd  +  dck  =  ~r  +  —r 
dx'  dc     ak 

~~rd(/>     r  cot<!>' 

In  Fig.  2,  the  points  a,  6,  and  k  are 
identical  with  the  points  similarly  lettered  in  Fig.  i.    The 


f _    1 -iff 

*      ? 

FIG.  2. 


Art.  4.]  EQUATIONS  IN  POLAR  CO-ORDINATES.  845 

expression  for  j-j-  may  be  at  once  written  from  Fig.  i.  There 
may,  then,  finally  be  written, 

du/     ^TI      TI  tan  0  du'  _       dcu 

=     ~~*~  ~ 


These  equations  will  give  the  expression  for  T^. 
The  value  of 

du'     dv'    dw' 

-+     + 


now  takes  the  following  form: 


dr)  doj      2p     ojtan 

rdr~        r 


The  last  two  terms  are  characteristic  of  the  spherical 
co-ordinates. 

The  eqs.  (20),  (21),  (22),  (n),  (12),  and  (13),  of  Art. 
i,  take  the  forms 


•  <« 


di,  do,  r,  tan       . 

r        '  '     ' 


846  ELASTICITY  OF  AMORPHOUS  SOLID  BODIES.        [Ch.  I. 

If  these  values  are  inserted  in  eqs.  (i),  (2),  and  (3),  the 
resulting  equations  will  be  applicable  to  isotropic  material 
only. 

As  in  the  preceding  article,  t  is  used  to  express  the  ratio 
between  direct  and  lateral  strains,  and  has  no  relation  what- 
ever to  the  co-ordinate  r. 

It  is-  interesting  and  important  to  observe  that  the  equa- 
tions of  motion  and  equilibrium  for  elastic  bodies  are  only 
special  cases  of  equations  which  are  entirely  independent  of 
the  nature  of  the  material,  of  equations,  in  fact,  which 
express  the  most  general  conditions  of  motion  or  equilibrium. 


CHAPTER  IT. 


THICK,    HOLLOW   CYLINDERS   AND   SPHERES,   AND 
TORSION. 

Art.  5.— Thick,  Hollow  Cylinders. 

IN  Fig.  i  is  represented  a  section,  taken  normal  to  its 
axis,  of  a  circular  cylinder  whose  walls  are  of  the  appreciable 
thickness  t.  Let  p  and  p^  represent  the  interior  and  exterior 
intensities  of  pressures,  respectively.  The  material  will  not 
be  stressed  with  uniform  intensity  throughout  the  thickness  t. 
Yet  if  that  thickness,  comparatively 
speaking,  is  small,  the  variation  will 
also  be  small;  or,  in  other  words, 
the  intensity  of  stress  throughout 
the  thickness  t  may  be  considered 
constant.  This  approximate  case 
will  first  be  considered. 

The  interior  intensity  p  will  be 
considered  greater  than  the  exterior 
pv  consequently  the  tendency  will 
be  toward  rupture  along  a  diametral  plane.  If,  at  the  same 
time,  the  ends  of  the  cylinder  are  taken  as  closed,  as  will  be 
done,  a  tendency  to  rupture  through  the  section  shown  in  the 
figure  will  exist. 

The  force  tending  to  produce  rupture  of  the  latter  kind 
will  be 

F  =  n(pr"-ps*) (i) 

847 


FIG.  i. 


848  THICK,  HOLLOW  CYLINDERS.  [Ch.  II. 

If  A7!  represents  the  intensity  of  stress  developed  by  this 
force, 


If  the  exterior  pressure  is  zero,  and  if  r*  is  nearly  equal  to 


In  this  same  approximate  case,  the  tendency  to  split  the 
cylinder  along  a  diametral  plane,  for  unit  of  length,  will  be 


If  Nf  is  the  intensity  of  stress  developed  by  F'9 


A7'  is  thus  seen  to  be  twice  as  great  as  Nl  when  p^  =  o.  If, 
therefore,  the  material  has  the  same  ultimate  resistance  in 
both  directions  the  cylinder  will  fail  longitudinally  when  the 
interior  intensity  is  only  half  great  enough  to  produce  trans- 
verse rupture,  the  thickness  being  assumed  to  be  very  small  and 
the  exterior  pressure  zero. 

N!  and  N'  are  tensile  stresses,  because  the  interior  pres- 
sure was  assumed  to  be  large  compared  with  the  exterior.  If 
the  opposite  assumption  were  made,  they  would  be  found  to 
be  compression,  while  the  general  forms  would  remain  ex- 
actly the  same. 


Art  5.]  THICK.  HOLLOW  CYLINDERS.  849 

The  preceding  formulas  are  too  loosely  approximate  for 
many  cases.  The  exact  treatment  requires  the  use  of  the 
general  equations  of  equilibrium,  and  the  forms  which  they 
take  in  Art.  3  are  particularly  convenient.  As  in  that  article, 
the  axis  of  x  will  be  taken  as  the  axis  of  the  cylinder. 

Since  all  external  pressure  is  uniform  in  intensity  and 
normal  in  direction,  no  shearing  stresses  will  exist  in  the 
material  of  the  cylinder.  This  condition  is  expressed  in  the 
notation  of  Art.  3  by  putting 

T$x  =••  Trx  =  Tr<p  =  o. 

Again  the  cylinder  will  be  considered  closed  at  the  ends, 
and  the  force  F,  eq.  (i),  will  be  assumed  to  develop  a  stress 
of  uniform  intensity  throughout  the  transverse  section 
shown  in  Fig.  i.  This  condition,  in  fact,  is  involved  in  that 
of  making  all  the  tangential  stresses  equal  to  zero. 

Since  this  case  is  that  of  equilibrium,  the  equations  (2). 
(3),  and  (4)  of  Art.  3  take  the  following  form,  after  neglect- 
ing XQt  R0,  and  00: 


dR    R- 


(7) 


These  equations  are  next  to  be  expressed  in  terms  of  the 
strains  u,  p,  and  w. 

In  consequence  of  the  manner  of  application  of  the  exter- 
nal forces,  all  movements  of  indefinitely  small  portions  of 


850  THICK,  HOLLOW  CYLINDERS.  [Ch.  II. 

the  material  will  be  along  the  radii  and  axis  of  the  cylinder. 
Hence 

u  will  be  independent  of  r  and  <j> ; 


The  rate  of  change,  therefore,  of  volume  will  be  (eq.  (6) 
of  Art.  3) 

du    dp    p 

u  =  -j — h~r~  +  ~~ (o) 

dx    dr     r  v  ' 


As  p  is  independent  of  x,  -j~  =;T^  ;  hence  if  the  value  of 

Nj  be  taken  from  eq.  (7)  of  Art.  3  and  put  in  eq.  (5)  of  this 
article, 

dN,      26*   d?u         d2u 
dx  ~i-2 


But  the  transverse  section  in  which  the  origin  is  located 
may  be  considered  fixed.  Consequently  if  x  =  o,  u=o  and 
thus  a'  =o.  The  expression  for  u  is  then  u  =ax. 

The  ratio  u  +  x  is  the  /  of  eq.  (i),  on  page  3,  while  the 
p  of  the  same  equation  is  simply  N1  of  eq.  (2),  given  above. 
Hence 


Art.  5.]  THICK,  HOLLOW  CYLINDERS.  851 

Again,    eq.  (8)  of  Art.  3,  in  connection  with  eqs.  (8) 

and  (6)  of  this,  gives 


-^)=o. 


:') 


r  dp-\-  p  dr  =d(pr)  =••  cr  dr. 

cr2  cr     b 

.'.  pr  =  —  +  b,  or    P  =  j  +  -.     •     •     •     (10) 

This  value  of  p  in  eqs.  (8)  and  (9)  of  Art.  3  will  give 


(12) 


At  the  interior  surface  R  must  be  equal  to  the  internal 
pressure,  and  at  the  exterior  surface  to  the  external  pressure. 
Or  since  negative  signs  indicate  compression, 


If  r  =  r' 
If  r  =  r, 


Either  of  these  equations  is  the  simple  result  of  applying 
eqs.  (13),  (14),  and.  (15)  to  the  present  case,  for  which 


cos  />=cos  r  =cos  TT=COS  /?  =  o, 
cos  q  =  cos  £  =  i  ,  and  P  =  —  p  or  —  pt. 


852  THICK.  HOLLOW  CYLINDERS.  [Ch.  II. 

Applying  eq.  (n)  to  the  two  surfaces, 

c      b 


b 


Subtracting  (14)  from  (13), 


Inserting  this  value  in  eq.  (13), 


i-2t 


The  general  expressions  of  R  and  N^,  freed  from  the 
arbitrary  constants  of  integration,  can  now  be  easily  written 
by  inserting  these  last  two  values  in  eqs.  (n)  and  (12).  By 
making  the  insertions  there  will  result 


» 


The  stress  N^  is  a  tension  directed  around  the  cylinder, 
and  has  been  called  "  hoop  tension."  Eq.  (16)  shows  that  the 
hoop  tension  will  be  greatest  at  the  interior  of  the  cylinder. 
An  expression  for  the  thickness,  t,  of  the  annulus  in  terms  of 
the  greatest  hoop  tension  (which  will  be  called  ti)  can  easily 
be  obtained  from  eq.  (16). 


Art.  6.]  TORSION  IN  EQUILIBRIUM.  853 

If  r  =/  in  that  equation, 


Eq.  (17)  will  enable  the  thickness  to  be  so  determined 
that  the  hoop  tension  shall  not  exceed  any  assigned  limit  h. 
If  pl  is  so  small  in  comparison  with  p  that  it  may  be  neg- 
lected, t  will  become 


:8) 


If  pl  is  greater  than  p,  N^  becomes  compression,  but 
the  equations  are  in  no  manner  changed. 

The  values  of  the  constants  b  and  c  may  easily  be  found 
from  the  two  equations  immediately  preceding  eq.  (15). 

It  is  interesting  to  notice  that  the  rate  of  change  of  vol- 
ume, 6,  is  equal  to  (a  +  c)  and  therefore  constant  for  all 
points. 

Art.  6. — Torsion  in  Equilibrium. 

The  formulas  to  be  deduced  in  this  article  are  those  first 
given  by  Saint- Venant,  and  established  in  substantially  the 
same  manner. 

It  will  in  all  cases,  except  that  of  the  final  result  for  a 
rectangular  cross-section,  be  convenient  to  use  those  equa- 
tions of  Art.  3  which  are  given  in  terms  of  semi -polar  co- 
ordinates. 


854 


TORSION   IN   EQUILIBRIUM. 


[Ch.  II. 


V 


-o — 


Let  Fig.  i  represent  a  cylindrical  piece  of  material,  with 
any  cross-section,  fixed  in  the  plane  ZY,  and  let  the  origin  of 

co-ordinates  be  taken  at  0.  Let 
it  be  twisted  also  by  a  couple 

P.ab=Pl, 

the  plane  of  which  is  parallel  to 
ZY.  The  material  will  thus  be 
subjected  to  no  bending,  but  to 
pure  torsion. 

The  axis  of  the  piece  is  sup- 
.  posed  to  be  parallel  to  the  axis 
of  X  as  well  as  the  axis  of  the 
couple.  Normal  sections  of  the 
piece,  originally  parallel  to  ZOF, 
will  not  remain  plane  after  tor- 
sion takes  place.  But  the  tendency  to  twist  any  elementary 
portion  of  the  piece  about  an  axis  passing  through  its  centre 
and  parallel  to  the  axis  of  X  will  be  very  small  compared 
with  the  tendency  to  twist  it  about  either  the  axis  of  r  or  <£; 
consequently  the  first  will  be  neglected.  In  the  notation 
of  Art.  3,  this  condition  is  equivalent  to  making  Tr^  =  o. 

As  the  piece  is  acted  upon  by  a  couple  only,  all  normal 
stresses  will  be  zero. 

Eqs.  (7),  (8),  (9),  and  (u)  of  Art.  3  then  become 


T    — 


(I) 


dp      dw    iv 


.     •     (3) 
(4; 


Art.  6.]  TORSION  IN  EQUILIBRIUM.  855 

After  introducing  the  values  of  Trx  and  7^,  from  eqs. 
(10)  and  (12)  of  Art.  3,  in  eqs.  (2),  (3),  and  (4)  of  the  same 
article,  at  the  same  time  making  the  external  forces  and 
second  members  of  those  equations  equal  to  zero,  and  bear- 
ing in  mind  the  conditions  given  above,  there  will  result 


dTrx    dT^x     Trx 

J  •  J    ;       ' 

dr       rd(>      r 


d2u      d2p 


(6) 


d2u 


Also  by  eq.  (6)  of  Art.  3, 


e+ 

dx     dr     ra<p     r 

The  cylindrical  piece  of  material  is  supposed  to  be  of 
such  length  that  the  portion  to  which  these  equations  apply 
is  not  affected  by  the  manner  of  application  of  the  couple. 
This  portion  is,  therefore,  twisted  uniformly  from  end  to 
end;  consequently  the  strain  u  will  not  vary  with  any 
change  in  x.  Hence 

du 


Eq.  (i)  then  shows  that  6  =  0.     This  was  to  be  antici- 
pated,  since  a  pure  shear  cannot  change  the  volume  or 


856  TORSION  IN  EQUILIBRIUM.  [Ch.  II. 

density.     Because  0  =  o,  eqs.  (2)  and  (3)  at  once  give 

dp      dw      p 

dr=^  +  r=0 (IO> 

As  the  torsion  is  uniform  throughout  the  portion  con- 
sidered, 

dp            dp  . 

-T-  =  O=— 7- (n) 

dx  r  dx 

Eq.  (n),  in  connection  with  eq.  (10),  gives 

d2w  (     . 

—1—7-7=0 (12) 

rdxd<f> 

Eqs.  (n)  and  (12),  in  connection  with  eq.  (10),  reduce 
eci-  (5)  to  the  following  form: 


d2u    ,d2u     du  d2/i 

=  0:=552"     dr 


d(  —} 
lu  ,       \dr) 


Both  terms  of  the  second  member  of  eq.  (6)  reduce  to 
zero  by  eqs.  (9)  and  (n),  and  give  no  new  condition.  The 
second  term  of  the  second  member  of  eq.  (7)  is  zero  by 
eq.  (9);  -the  remaining  term  therefore  gives 

d2w 


As  the  stress  is  all  shearing,  p  will  not  vary  with  <f>. 
Hence 


Art-  6-]  TORSION  IN  EQUILIBRIUM.  857 

Eqs.  (10),   (n),  and  (15)  show  that  p-o,  and  reduce 
eq.   (4)  to 

dw    w 

-     =  0.     .'    ,     .     .     ./.     (I6) 


Eq.  (10)  now  becomes          =o,  and  shows  that  w  does 


not  contain  0;  while  eq.  (14)  shows  that  w  does  not  con- 
tain x2  or  any  higher  power  of  x.  The  strain  w,  in  connec- 
tion with  these  conditions,  is  to  be  so  determined  as  to  sat- 
isfy eq.  (16). 

If  a  is  a  constant,  the  following  form  fulfils  all  condi- 
tions : 

w=--arx  ........     (17) 

Eq.  (17)  shows  that  the  strain  w,  in  the  direction  of  0, 
i.e.,  the  angular  strain  at  any  point,  varies  directly  as  the  dis- 
tance from  the  axis  of  X,  and  as  the  distance  from  the  origin 
measured  along  that  axis.  This  is  a  direct  consequence  of 
making  7^  =  0. 

The  quantity  a  is  evidently  the  angle  of  torsion,  or  the 
angle  through  which  one  end  of  a  unit  of  fibre,  situated  at 
unit's  distance  from  the  axis,  is  twisted  ;  for  if 


An  equation  of  condition  relative  to  the  exterior  surface 
of  the  twisted  piece  yet  remains  to  be  determined  ;  and  that 
is  to  be  based  on  the  supposition  that  no  external  force  what- 
ever acts  on  the  outer  surface  of  the  piece.  In  eqs.  (13), 
(14),  and  (15)  of  Art.  2,  consequently,  P  =  o.  The  conditions 
of  the  problem  also  make  all  the  stresses  except 

T  =  T       and     T 


858  TORSION  IN  EQUILIBRIUM.  [Ch.  II. 

equal  to  zero,  while  .the  cylindrical  character  of  the  piece 
makes 

£  =  90°;     .*.  cos  p=o. 

If  cos  t  be  written  for  cos  r, 

cos  t  =  sin  q. 
Eq.  (13),  just  cited,  then  gives 

Txr  cos  q  +  T<f,x  sin  q  =  o (18) 

But  since  p  =  o  and  w  =  arx, 

Txr=G^  (19) 

dr 

and 


Eq.  (18)  now  becomes 
rf* 


dr, 

,       .     .     .     (21) 


in  which  r0  is  the  value  of  r  for  the  perimeter  of  any  normal 
section. 

Eqs.  (13)  and  (21)  are  all  that  are  necessary  and  all  that 
exist  for  the  determination  of  the  strain  u.  Eq.  (13)  must 
be  fulfilled  at  all  points  in  the  interior  of  the  twisted  piece, 
while  eq.  (2  1  )  must  at  the  same  time  hold  true  at  all  points 
of  the  exterior  surface. 


Art-  6.]  TORSION  IN  EQUILIBRIUM.  859 

After  u  is  determined,  Txr  and  Tx<f>  at  once  result  from 
eqs.  (19)  and  (20).  The  resisting  moment  of  torsion  then 
becomes 


In  this  equation  IP  =  J  Jr2  .  r  dc/>  dr  is  the  polar  moment  of 

inertia  of  the  normal  section  of  the  piece  about  the  axis  of 
X,  and  the  double  integral  is  to  be  extended  over  the  whole 
section. 

According  to  the  old  or  common  theory  of  torsion 

M=GaIP. 

The  third  member  of  eq.  (22)  shows,  however,  that  such  an 
expression  is  not  correct  unless  u  is  equal  to  zero ;  i.e.,  unless 
all  normal  sections  remain  plane  while  the  piece  is  subjected 
to  torsion.  It  will  be  seen  that  this  is  true  for  a  circular  sec- 
tion only. 

It  may  sometimes  be  convenient  to  put  eq.  (22)  in  the 
following  form: 

r  r      du  .  f 

M  =G  I     I  rdr.-Tjd(j)-{-GaIp  =  GI  u.rdr  +  Ga!p.     (23) 

In  this  equation  u  is  to  be  considered  as 

/*  du 
-r—d^j 
dfp 

while  the  remaining  integration  in  r  is  to  be  so  made  that 
the  whole  section  shall  be  covered. 


86o  TORSION  IN  EQUILIBRIUM.  [Ch.  II. 

The  preceding  analysis  shows  that  the  old  or  common 
theory  of  torsion  is  correct  in  its  expression  for  torsive 
strain,  as  it  is  identical  with  eq.  (17)  of  Art.  6,  i.e., 


but  it  will  be  seen  later  that  the  remaining  formulae  of  the 
common  theory  are  incorrect  for  all  shapes  of  cross-section 
except  the  circle.  Fortunately  the  torsion  members  prin- 
cipally used  in  engineering  practice  are  shafts  of  circular 
section. 

Equations  of  Condition  in  Rectangular  Co-ordinates. 

In  the  case  of  a  rectangular  normal  section,  the  analysis 
is  somewhat  simplified  by  taking  some  of  the  quantities 
used  in  terms  of  rectangular  co-ordinates. 

In  the  notation  of  Art.  2  all  stresses  will  be  zero  except 
T3  and  T2.  Hence  eqs.  (10),  (n),  and  (12)  of  that  article 
reduce  to 

dT,    dT2 
dy       dz  ~°-' 


d% 


doo 


=o; 


=  o. 


The  strains  in  the  directions  of  x,  y,  and  z  are,  respec- 
tively, 11,  v,  and  w.  Introducing  the  values  of  Ts  and  T2 
in  the  equations  above,  in  terms  of  these  strains,  from 
eqs.  (n)  and  (13)  of  Art.  i,  and  then  doing  the  same  in 
reference  to  the  conditions, 


Art.  6.]  TORSION  IN  EQUILIBRIUM.  86  1 

the  following  equations  will  result: 


57+5?=°'  ......    (26) 

dv    dw 


The  operations  by  which  these  results  are  reached  are 
identical  with  those  used  above  in  connection  with  semi- 
polar  co-ordinates,  and  need  not  be  repeated. 

Eq.  (27)  is  satisfied  by  taking 

v  =     axz  ; 
w  =  '—  axy  ; 

in  which  a  is  the  angle  of  torsion,  as  before. 
Eqs.  (n)  and  (13)  of  Art.  5  then  give 

dv        -  ,  c 

(28) 


du    dw\          du 


The  element  of  a  normal  section  is  dz  dy.     Hence  the 
moment  of  torsion  is 


/.  M=Gf(zudz-yudy)+GaIP  .....     (31) 


862  TORSION  IN  EQUILIBRIUM.  [Ch.  II. 

is  the  polar  moment  of  inertia  of  any  section  about  the 
axis  of  X. 

The  integrals  are  to  be  extended  over  the  whole  section ; 
hence,  in  eq.  (31),  zu  dz  is  to  be  taken  as 


zdz.   I        y-dy 
J -y0   dy  * 

and  yu  dy  as 


in  which  expressions  y0  and  00  are  general  co-ordinates  of 
the  perimeter  of  the  normal  section. 

Eq.  (26)  is  identical  with  eq.  (13),  and  can  be  derived 
from  it,  through  a  change  in  the  independent  variables,  by 
the  aid  of  the  relations 

>     and        =rsin<. 


Solutions  of  Eqs.  (13)  -and  (21). 

It  has  been  shown  that  the  function  u,  which  represents 
the  strain  parallel  to  the  axis  of  the  piece,  must  satisfy 
eq.  (13)  [or  eq.  (26)]  for  all  points  of  any  normal  section, 
and  eq.  (21)  (or  a  corresponding  one  in  rectangular  co- 
ordinates) at  all  points  of  the  perimeter  ;  and  those  two  are 
the  only  conditions  to  be  satisfied. 

It  is  shown  by  the  ordinary  operations  of  the  calculus 
that  an  indefinite  number  of  functions  u,  of  r  and  0,  will 
satisfy  eq.  (13)  ;  and,  of  these,  that  some  are  algebraic  and 
some  transcendental. 

It  is  ftirther  shown  that  the  various  functions  u  which 
satisfy  both  eqs.  (13)  and  (21)  differ  only  by  constants. 


Art.  6.]  TORSION  IN  EQUILIBRIUM.  863 

If  u  is  first  supposed  to  be  algebraic  in  character,  and  if 
cv  c2,  c3,  etc.,  represent  constant  coefficients,  the  following 
general  function  will  satisfy  eq.  (13): 


_  sn  (>  +  cr    sn  2<-K       sn 


and  the  following  equation,  which  is  supposed  to  belong  to 
the  perimeter  of  a  normal  section  only,  will  be  found  to 
satisfy  eq.  (21) : 

r2 

—  +  c.r  cos  (/)  +  c2r2  cos  2<£  +  £3r3  cos 

2 


—  cflr  sin  <p  —  cf2r2  sin  2  <£  —  c'srs  sin  3 (f>  —  .  .  .  =  C.     (33) 

C  is  a  constant  which  changes  only  with  the  form  of 
section. 

T,  du  du          ,        .  dr0 

If  -r-  and      *  ,  be  found  from  eq.  (32),  while  — -7-7  be 

taken  from  eq.  (33),  and  if  these  quantities  be  then  intro- 
duced in  eq.  (21),  it  will  be  found  that  that  equation  is 
satisfied. 

The  only  form  of  transcendental  function  needed, 
among  those  to  which  the  integration  of  eq.  (13)  or  eq.  (26) 
leads,  will  be  given  in  connection  with  the  consideration  of 
pieces  with  rectangular  section,  where  it  will  be  used. 

Elliptical  Section  about  its  Centre. 

Let  a  cylindrical  piece  of -material  with  elliptical  normal 
section  be  taken,  and  let  a  be  the  semi-major  and  b  the 
semi-minor  axis,  while  the  angle  <£  is  measured  from  a 
with  the  centre  of  the  ellipse  as  the  origin  of  co-ordinates, 
since  the  C37linder  will  be  twisted  about  its  own  axis.  The 


864  TORSION  IN  EQUILIBRIUM.  [Ch.  II. 

polar  equation  of  the  elliptical  perimeter  may  take  the 
following  shape: 


""  +  """  '  ~~r~TT?  COS  2  0  = 

2      2    a2  +  62  Y 


By  a  comparison  of  eqs.  (33)  and  (34),  it  is  seen  that 
c2=   (a*~?b9\     and     C=  ?    ,a, 

and  that  all  the  other  constants  are  zero.     Hence  eq.  (32) 
gives 

62-a2  a 

U  =  a2(a*  +  b2)     Sm  2 ^  =  ^r  sm  2 ^'    '     '     (35) 


The  quantity  represented  by  /  is  evident. 
By  eqs.  (19)  and  (20) 


b2  —  a2 

r  sin  2(56;  .....     (36) 


H.  .     .     .     (37) 


Since    °'  °       =^^4,  A  being  the  area  of  the  ellipse,  or 

nab,  the  second  member  of  eq.  (22),  by  the  aid  of  eq.  (37), 
may  take  the  form 

M  =Ga  I  dd>  I       -2 — r-S  cos  2 


Art.  6.]  TORSION  IN  EQUILIBRIUM.  865 

Then  using  eq.  (34), 


(38) 


If  Ip  is  the  polar  moment  of  inertia  of  the  ellipse  (i.e., 
about  an  axis  normal  to  its  plane  and  passing  through  its 
centre),  so  that 


then 

M =Ga — 55-7-. 
47r"L> 


Using  /  in  the  manner  shown  in  eq.  (35),  the  resultant 
shear  at  any  point  becomes,  by  eq.  (24), 


+  2    cos 
dT 


gives 

siri2<£  =  o,     or     ^  =  90°    or    o°. 

Since  /  is  negative,  T  will  evidently  take  its  maximum 
when  <j>  has  such  a  value  that  2}  cos  2^  is  positive,  or  <j> 
must  be  90°. 

Hence  the  greatest  intensity  of  shear  will  be  found  some- 
where along  the  minor  axis.  But  the  preceding  expression 
shows  that  T  varies  directly  as  the  distance  from  the  centre. 
Hence  the  greatest  intensity  of  shear  is  found  at  the  extremities 
of  the  minor  axis. 


866  TORSION  IN  EQUILIBRIUM. 

Making  <£  =  90°  and  r  =  b  in  the  value  of  T, 

2a2b 


[Ch.  II. 


(40) 


Taking  Ga  from  eq.  (40)  and  inserting  it  in  eq.  (38), 


(41) 


in  which 


or  the  moment  of  inertia  of  the  section  about  the  major  axis. 

Equilateral  Triangle  about  its  Centre  of  Gravity. 

This  case  is  that  of  a  cylindrical  piece  whose  normal  cross- 
section  is  an  equilateral  triangle,  and  the  torsion  will  be  sup- 
posed about  an  axis  passing  through 
the  centres  of  gravity  of  the  different 
normal  sections.  The  cross-section  is 
represented  in  Fig.  3,  G  being  the  H 
centre  of  gravity  as  well  as  the  origin 
of  co-ordinates. 

Let  GH  =  \GD  =a.   Then  from  the    { 
known  properties  of  such  a  triangle,  FIG.  3. 

FD  =  DB  --=  BF  =  2a\/3. 

2a  —  r  cos  d> 
Hence  the  equation  for  DB  is  ;  r  sin  </)  --  ;=  —  z  =  o  . 


Hence  the  equation  for  BF  is  ; 


r  cos  <£  f  a  =  o  . 


Hence  the  equation  for  FD  is  ;  r  sin  <f>  +  -    —,-=  —  -  =  o  . 


Art.  6.]  TORSION  IN  EQUILIBRIUM.  867 

Taking  the  product  of  these  three  equations  and  reduc- 
ing, there  will  result  for  the  equation  to  the  perimeter 


—  </>=  -  ......       (42) 

2      6a  3 


Comparing  this  equation  with  eq.  (33) 


i  20, 

T~     and     C  =  — 
6a  3 


j        r-       2a 

cs=——     and     C  = — • 


Hence 


=0.6  GaIP  =  i.S  G'aaV;  .     (46) 


since  7^,=  polar  moment  of  inertia  =  3«4\/3. 


6a 
And  by  eqs.  (19)  and  (20) 

Txr  =  -Ga-    ^-;  .....     (44) 


-  .....     (45) 

Eq.  (22)  then  gives 

M=CaIt-Gaf  r- 
t/   «y 


868  TORSION  IN  EQUILIBRIUM.  Ch.  II. 

By  eq.  (24) 


<t>      r' 
a"    +^<      •     •     •     (47) 


.".  ~JT=O     gives     sm30=o, 

or 

0=o°,  60°,  120°,  180°,  240°,  300°,  or  360°. 

The  values  o°,  120°,  240°,  and  360°  make 
cos  30=  +i; 

hence,  for  a  given  value  of  r,  these  make  T  a  minimum.  The 
values  60°,  1 80°,  and  300°  make, 


hence,  for  a  given  value  of  r,  these  make  T  a  maximum. 
Putting  cos  30  =  —  i  in  eq.  (47), 


(48) 


This  value  will  be  the  greatest  possible  when  r  is  the 
greatest.  But  $  =  60°,  180°,  and  300°  correspond  to  the  nor- 
mal a  dropped  on  each  of  the  three  sides  of  the  triangle 
from  G.  Hence  r  =  a,  in  eq.  (48),  gives  the  greatest  intensity 
of  shear  T  ,  or 

yn  * 


(49) 


Art.  6.]  TORSION  IN  EQUILIBRIUM.  869 

Or  the  greatest  intensity  of  shear  exists  at  the  middle  point 
of  each  side.  Those  points  are  the  nearest  of  all,  in  the 
perimeter,  to  the  axis  of  torsion. 

The  value  of  Ga,  from  eq.  (49),  inserted  in  eq.  (46), 
gives 

Z3T 

' 


in  which  /=  side  of  section  =  2<n/3. 


Rectangular  Section  about  an  Axis  passing  through  its 
Centre  of  Gravity. 

In  this  case  it  will  be  necessary  to  consider  one  of  the 
transcendental  forms  to  which  the  integration  of  eq.  (13) 
[or  (26)]  leads;  for  if  the  polar  equation  to  the  perimeter  be 
formed,  as  was  done  in  the  preceding  case,  it  will  be  found 
to  contain  r4,  to  which  no  term  in  eq.  (33)  corresponds. 

If  e  is  the  base  of  the  Napierian  system  of  logarithms 
(numerically  £  =  2.71828,  nearly)  and  A  any  constant  what- 
ever, it  is  known  that  the  general  integral  of  the  partial 
differential  eq.  (13)  may  be  expressed  as  follows: 


But  the  second  member  of  this  equation  is  evidently 
equal  to  zero  if 


or 


870  TORSION  IN  EQUILIBRIUM.  [Ch.  II. 

These  relations  make  it  necessary  that  neither  n  or  n'  shall 
be  imaginary. 

It  will  hereafter  be  convenient  to  use  the  following  no- 
tation for  hyperbolic  sines,  cosines,  and  tangents : 


By  the  use  of  Euler's  exponential  formula,  as  is  well 
known,  and  remembering  that  n'2=—n2,  eq.  (51)  may  be 
put  in  the  following  form : 

u  =  Ienr  cos *  [A n  sin  (nr  sin  <j>)  +A'H  cos  (nr  sin  $)], 

in  which  the  sign  of  summation  is  to  be  extended  to  all  pos- 
sible values  of  An  and  A'n.  At  the  centre  of  any  section  for 
which  r  is  zero,  u  must  be  zero  also,  for  the  axis  of  the  piece 
is  not  shortened.  This  condition  requires  that  A'n  =  o;  u 
then  becomes 

u  =  Ienr  cos  *  A  n  sin  (nr  sin  <£) . 

The  subsequent  analysis  will  be  simplified  by  introduc- 
ing the  form  of  the  hyperbolic  sine,  and  this  may  be  done 
by  adding  and  subtracting  the  same  quantity  to  that  al- 
ready under  the  sign  of  summation,  in  such  a  manner  that 

u  =  2\A  n  sin  (nr  sin  <p) .  sih  (nr  cos  <£) 

+  %A  „  sin  (nr  sin  <£)  e~nr  cos #].     (52) 

Now  if  the  product 

sin  (nr  sin  <f>) 


Art.  6.j 


TORSION  IN  EQUILIBRIUM. 


87I 


be  developed  in  a  series  and  multiplied  by  Ant  one  term  will 
consist  of  the  quantity 

—  r2  sin  (j>  cos  <£ 
multiplied  by  a  constant,  and  if 

IAn  sin  (nr  sin  <j>)  e~nrcos^ 
be  replaced  by  simply, 

—  ar2  sin  </>  cos  <£, 

all  the  conditions  of  the  problem  will  be  found  to  be  satis- 
fied.   This  is  equivalent  to  putting 

—  ar2  sin  <£  cos  <j> 

for  a  general  function  of  r  sin  (f>  and  r  cos  <£.  This  change  will 
give  the  following  form  to  «,  first  used  by  Sain t-Ven ant : 

u  •=  I A  n  sin  (nr  sin  0) .  sih  (nr  cos  <£)  —  ar2  sin  <j>  cos  <£.     (53) 

Fig.  4  represents  the  cross-section  with  C  as  the  origin  of 
co-.ordinates  and  axis.     The  angle  (j>  is  measured  positively 


1 

^'' 

V 

t 

1 

1 

1 
1 

1 

3 

FIG.  4. 

D 

from  CN  toward  CH.   At  the  points  N,  H,  K,  and  L,  in  the 


872  TORSION  IN  EQUILIBRIUM.  [Ch.  II. 

equation  to  the  perimeter,  dr0  will  be  zero.     Hence  at  those 
points,  by  eq.  (21), 

-i-  =  I[A  n  sin  (nr  sin  <j>) .  n  cos  <j> .  coh  (nr  cos  <£) 

.  n  sin  </> .  cos  (nr  sin  <£) .  sih  (nr  cos  </>)] 
—  2ar  sin  (>  cos  <   =o. 


At  the  points  under  consideration  <£  has  the  values  o°, 
90°,  1 80°,  270°,  and  360°.  At  the  points  N  and  K,  <j>  =0°  or 
1 80°;  hence  sin  </>  =  o,  and  both  terms  of  the  second  mem- 
ber of  -j-  reduce  to  zero,  whatever  may  be  the  value  of  n. 

But  at  H  and  L,  <£  =  90°  and  270° ;  hence  sin  <j>  =  +  i  or  —  i 
and  cos  <£  =  o. 

In  order,  then,  that  -r*  =o  at  H  and  L,  these  must  obtain : 

cos  nr  =  cos  (  —  nr)=  o. 
If  HL=c  and  KN  =  b,  then 


nc 
cos  - 

2 


=COS(~T)==O (54) 


If  the  signification  of  n  be  now  somewhat  changed  so  as 
to  represent  all  possible  whole  numbers  between  o  and  oo , 
eq.  (54)  will  be  satisfied  by  writing 


2n—  i 

7T 

c 


Art.  6.]  TORSION  IN  EQUILIBRIUM.  873 

for  n  in  that  equation.     Eq.  (53)  will  then  become 

<*>  .       /2tt—  I  .          \  /2H—  I 

u  =  2An  sin  I  -  nr  sin  <£  )  .  sih  ( 

1  \       C  /  \ 


nr  cos 


\       C 

?     .    .     .     (55) 

The  quantity  A  n  yet  remains  to  be  determined  by  the 
aid  of  eq.  (21),  which  expresses  the  condition  existing  at 
the  perimeter  of  any  section. 

Now,  for  the  portion  BN  of  the  perimeter, 

b 

r  cos  <p  =—  > 


and  -— A  will  be  the  tangent  of  (—<£),  or 


dr°   =-  tan  (-</>)=  tan  <£. 


Hence  eq.  (21)  becomes 

!u 

^ 

=  tan  <j>, (56) 


du 

dr 


du 


or 

dw  du     . 

ar  sin  (   =       cos  ^~          sm  <^' 


Substituting  from  eq.  (55),  then  making 
r  cos  <£=-, 


«  2H—I  .      2H—I     ,          .          2H—I 

r  sin     =  J.4    .  -          «  .  coh-        «fc   .  sm          —  *r  sm 


§74  TORSION  IN  EQUILIBRIUM.  [Ch.  II. 

If  r  sin  cj)  be  represented  by  the  rectangular  co-ordinate 
y,  and  another  quantity  by  H,  the  above  equation  may  be 
written 


TT        . 

.  .  .   -//sm 


c 

2H— 


If  both  sides  of  this  equation  be  multiplied  by 

-I        \       , 

—nyJ.dy, 


2H- 

sin 


and  if  the  integral  then  be  taken  between  the  limits  o 
and  -,  it  is  known  from  the  integral  calculus  that  all  terms 
except  the  nth  will  disappear,  and  that 


c_ 
2- 

sm 


Completing  these  simple  integrations, 

#«  = 

Hence 


/      T     \  II         J.  /~-  .* 

/l      _  v  ~ T  /       c      4          

^     M  —  7  \  '     2  "         •   /  \ 


Art.  6.] 


TORSION  IN  EQUILIBRIUM. 


875 


If  this  value  of  A  n  be  put  in  eq.  (55),  and  if  rectangular 
co-ordinates 

=r$m(>     and     2= 


be  introduced,  that  equation  will  become 


3  • 

/2\3  ac^S 
W  ' 


.Slh 


•      (57) 


This  value  of  u  placed  in  eq.  (31)  will  enable  the  moment 
of  torsion  to  be  at  once  written. 

The  limits  +yQ  and  —  y0  are  +-  and  —  ,  and  the  limits 


0  and  —  ZQ  are  -f-  and  —  -.     Hence 


H- 


=  abc 


n, 

2  Slh 


b-tfi  f 


\ 


1 


2H—  i)3  COh 


2^,1,) 

2C  /  J 


=  Q,  for  brevity; 


=  abc 


/  vi        (-i)71"1^  sih(— — -Kb)  .sin  (-^—-ny) 

-(-}  -2 \    2c      I        \    c      'i  r 

\        2(7  /  J 


6.]  TORSION  IN  EQUILIBRIUM. 

For  the  next  integration 


[Ch.  II. 


/-\ 
2  Qzdz  = 
b 


abc 


12 


2bc 


.coh: 


/  \    .. 

(2H—  I)n  2C 


*^*ti 

2C  / 


(2W-l)3 


f/Rydy  = 


12     \xj  b    x 


2C 


—  i)3  coh 


(Szirt) 

\      2C  /    J 


Thus  the  integrations  indicated  in  eq.    (31)   are  com 
pleted.     Hence 


M  = 
Remembering  that 


^=6^ 


M 


tah  ( Tib] 

_  *£.*&*?* L_  64C4^,  \       2C  / 

=  CzaL6    "     "    ^      t(2"-l)4  7T5    t          (2W-I)5         J' 


(58) 


But  it  is  known  that 


2  7T1 


T    (2W—  l)4       1.2.3     25 


Art.  6.]  TORSION  IN  EQUILIBRIUM.  877 

Hence  eq.  (58)  becomes 


r 

|_ 


tah 

.     (S9) 


Since 


(i  —  tah  TT      i—  tah  37T-    i—  tab  5^  \ 

~        ~Y~        ~?~  7 


tah  Ti     tah    ;:     tah 


i  35  5s 

and  since 

64 


—  =  0.209137, 


and  remembering  that 

-f.       /  \     f  / 

2(^rJ  =  i+£5  +  4  +  •  •  •  -(*— i 

*  \ .   '         /  o        o  \ 

eq.  (59)  becomes 

•  3|    -  —  0.210083^- 

L3 

0.209137^1-       -+ 


,--:<*  ....    ,- 

i  —  tah  -       i  —  tah  — • 

2C 


Eq.  (60)  gives  the  value  of  the  moment  of  torsion  of  a 
rectangular  bar  of  material. 


878  TORSION  IN  EQUILIBRIUM.  [Ch.  II. 

If  z  had  been  taken  parallel  to  b,  and  y  parallel  to  c,  a 
moment  of  equal  value  would  have  been  found,  which  can 
be  at  once  written  from  eq.  (60)  by  writing  b  for  c  and  c  for  6. 

That  moment  will  be 

M --=Gacb*\      —0.210083- 


,  , 

i-tah-r     i-tah-r- 


+  0.209137-!-  -+  --  -5  --  +.../    |.     (6i) 

*". 


Eq.  (60)  should  be  used  when  b  is  greater  than  c,  and  eq. 
(61)  when  c  is  greater  than  6,  because  the  series  in  the  paren- 
theses are  then  very  rapidly  converging,  and  not  diverging. 
It  will  never  be  necessary  to  take  more  than  three  or  four 
terms  and  one,  only,  will  ordinarily  be  sufficient.  The  follow- 
ing are  the  values  of 


/  1     "7T\ 

(x-tehT) 


for  a  few  values  of  n: 


1 1  — tah  —  j  =0.083  :  °-°°373  :  0.000162  :  0.000007 ; 


Square  Section. 
If  c=b  either  eq.  (60)  or.eq.  (61)  gives 

M=Gab4\     —0.2101+0.209(1—  tah-J    ; 

44 

.*.   M  =0.1406  Gab4  =  Ga  -    — j-,        .     .     (62) 

42.7  J-  + 


Art.  6.]  TORSION  IN  EQUILIBRIUM.  879 

in  which  A  is  the  area  (  =  62)  and  IP  is  the  polar  moment  of 

/     b*\ 
inertia      =. 


Rectangle  in  which  b  =  2C. 
If  b  =  2C,  eq.  (60)  gives 


**Ga*2C*  —  0.105  +  0.1046  (i  —  tah  n)    ; 


44 

;  -  .     .-   (63) 


42 


in  which  A  is  the  area  (  =  2C2)  and  7^,  =  polar  moment  of  in- 
ertia 


12  6 

Rectangle  in  which  b  =  4.0. 
If  b  =  4C,  eq.  (60)  then  gives 


!  —  0.0525  J  =1.123  Gac4', 

\o  / 

.     .     .    -.     .     .     .     .     (64) 


.        -^--r,         .     .     .     .     .     .     . 

40-2  IP 

in  which  A  =area  =  4<:2  and  IP  =  polar  moment  of  inertia 


12 


88o  TORSION  IN  EQUILIBRIUM.  [Ch.  II. 

If  b  is  greater  than  20,  it  will  be  sufficiently  near  for  all 
ordinary  purposes  to  write 


M--=Ga—  li-o.6ir 


Greatest  Intensity  of  Shear. 

There  yet  remains  to  be  determined  the  greatest  inten- 
sity of  shear  at  any  point  in  a  section,  and  in  searching  for 
this  quantity  it  will  be  convenient  to  use  eqs.  (28)  and  (29). 

It  will  also  be  well  to  observe  that  by  changing  z  to  y, 
y  to  —  z,  c  to  6,  and  b  to  c,  in  eq.  (57),  there  may  be  at  once 
written 


.        2H  — 


(2H—  l)3Coh 


2b 


.  (66) 


This  amounts  to  turning  the  co-ordinate  axes  90°. 
Since  the  resultant  shear  at  any  point  is 


it  will  be  necessary  to  seek  the  maximum  of 
du        \2     Idu  T2 


The  two  following  equations  will  then  give  the  points 
desired  : 


Art.  6.]  TORSION  IN  EQUILIBRIUM.  88 1 

/du         \d*u     /du 

=(^+az)w+(^- 
d/ir\ 

\G2/      /du         \/  d*u         \      /du         \d*u 
_\— ^  =/       +a2j)(        -+a\  +  (      _ay\        =0,    (68) 

dz         \dy         /  \dzdy       )      \dz        ^Jdz2 

It  is  unnecessary  to  reproduce  the  complete  substitu- 
tions in  these  two  equations,  but  such  operations  show  that 
the  points  of  maximum  values  of  T  are  at  the  middle  points  of 
the  sides  of  the  rectangular  sections,  omitting  the  evident  fact 
that  r  =  o  at  the  centre.  It  will  also  be  found  that  the  great- 
est intensity  of  shear  will  exist  at  the  middle  points  of  the 
greater  sides. 

This  result  may  be  reached  independent  of  any  analytical 
test,  by  bearing  in  mind  that  an  elongated  ellipse  closely 
approximates  a  rectangular  section,  and  it  has  already  been 
shown  that  the  greatest  intensity  in  an  elliptical  section  is 
found  at  the  extremities  of  the  smaller  axis. 

By  the  aid  of  eqs.  (28),  (29),  (57),  and  (66),  it  will  also 
be  found  that  T3  =  o  at  the  extremities  of  the  diameter  c, 
and  T2  =  o  at  the  extremities  of  the  diameter  b.  The  maxi- 
mum value  of  T  will  then  be 


By  the  use  of  eq.  (57) 


du 

~  —  ay  — 

dz 


(2n—  i)2  coh  (  --  no  1 


882  TORSION  IN  EQUILIBRIUM.  [Ch.  II. 

Putting  z=o  and  y  =—  in  this  equation,  there  will  result 


il-GoTi-Al rl--     (70) 


If  b  is  greater  than  c  the  series  appearing  in  this  equation 
is  very  rapidly  convergent,  and  it  will  never  be  necessary  to 
use  more  than  two  or  three  terms  if  the  section  is  square,  and 
if  b  is  four  or  five  times  c  there  may  be  written 


(71) 


Square  Section. 


Making  b  =c  in  eq.  (70),  and  making  n  =  i,  2,  and  3  (i.e., 
taking  three  terms  of  the  series),  there  will  result 


0.676  Gac;     :.'Ga  =  1.48— -. 

c 


Inserting  this  value  in  eq.  (62), 


M  <     , 

-p,      ....     (73) 


in  which 


T  A  c 

1= —     and     a=-=-, 

12  22 


Art.  6.]  TORSION  IN  EQUILIBRIUM.  883 

Rectangular  Section;  b  =  2C. 

Making  b  =  2C  in  eq.  (70),  and  making  n  =  i,  only,  there 
will  result 

r 

T    =o.9$Gac',     .'.  Ga  =  i.o&-*. 

c 


Inserting  this  value  in  eq.  (63), 


-  •     •     •     •     (74) 


M         M 
a^23,      ....     (75) 


in  which 


r     be3     c4         ^  c 

I  —  —  =-2~     and     a  =  — . 

126  2 


Rectangular  Section;  b=4C. 
Making  b  =  ^c  in  eq.  (70),  and  making  w  =  i,  only, 

Tm  =0.997    ' 


Inserting  this  value  in  eq.   (64), 

M  =  i.  126  r37m  =  1.69^.      •     •  >•     (76) 

.....     (77) 


in  which 


=  —  =—     and     a=-. 
123  2 


884  TORSION  IN  EQUILIBRIUM.  [Ch.  II. 


Circular  Section  about  its  Centre. 

The  torsion  of  a  circular  cylinder  furnishes  the  simplest 
example  of  all. 

If  r0  is  the  radius  of  the  circular  section,  the  polar  equa- 
tion of  that  section  is 


—  =  C  (constant). 


Comparing  this  equation  with  eq.  (33),  it  is  seen  that 

By  eq.  (32)  this  gives  u  =  o.     Hence  all  sections  remain 
plane  during  torsion. 

Eqs.   (19)  and  (20)  then  give 

Txr  =  o     and     Tx^=Gar (78) 

Eq.  (23)  gives  for  the  moment  of  torsion 

M=GaIpt (79) 

or 

in  which  equation  A  is  the  area  of  the  section  and 

4 

P         2 


Art.  6.]  TORSION  IN  EQUILIBRIUM.  885 

The  greatest  intensity  of  shear  in  the  section  will  be  ob- 
tained by  making  r  =  r0  in  eq.  (78),  or 


;    .-.C«=-^.   .    .    .    .    (81) 
Eq.  (80)  then  becomes 

Jl/  =  o.5  7rr0srm  =  2 — « (82) 

ro 

A/         M 
.'.  rm=o.64— 3=o.5yr0, (83) 

in  which  /  =  — — • 

4 

It  is  thus  seen  that  the  circular  section  is  the  only  one 
treated  which  remains  plane  during  torsion. 

General  Observations. 

The  preceding  examples  will  sufficiently  exemplify  the 
method  to  be  followed  in  any  case.  Some  general  conclu- 
sions, however,  may  be  drawn  from  a  consideration  of 

eq.  (33). 

If  the  perimeter  is  symmetrical  about  the  line  from 
which  <j)  is  measured,  then  r  must  be  the  same  for  +  <f>  and 
—  $ ;  hence 

c/  =  c2f  =  cs'  =  .  .  .    =  o. 

If  the  perimeter  is  symmetrical  about  a  line  at  right 
angles  to  the  zero  position  of  r,  then  r  must  be  the  same  for 

(h  =  oo  -f~  <b     and     QO  —  <p  \ 


886  TORSION AL  OSCILLATIONS.  [Ch.  II. 

hence 

^=^3  =  ^5.  ••     =cj=cf=cj-  ....  =o. 

In  connection  with  the  first  of  these  sets  of  results, 
eq.  (32)  shows  that  every  axis  of  symmetry  of  sections  repre- 
sented by  eq.  (33)  will  not  be  moved  from  its  original  position 
by  torsion. 

If  the  section  has  two  axes  of  symmetry  passing  through 
the  origin  of  co-ordinates,  then  will  all  the  above  constants 
be  zero,  and  its  equation  will  become 


cos  2 <j>  +  c4r4  cos  4^  +  c^r*  cos 


Art.  7.  —  Torsional  Oscillations  of  Circular  Cylinders. 

Two  cases  of  torsi  onal  oscillations  will  be  considered, 
in  the  first  of  which  the  cylindrical  body  twisted  is  sup- 
posed to  be  the  only  one  in  motion.  In  the  second  case, 
however,  the  mass  of  the  twisted  body  will  be  neglected, 
and  the  motion  of  a  heavy  body,  attached  to  its  free  end, 
will  be  considered.  In  both  cases  the  section  of  the  cylin- 
der will  be  considered  circular. 

Since  these  cases  are  those  of  motion,  the  internal 
stresses  are  not,  in  general,  in  equilibrium;  hence  equations 
of  motion  must  be  used,  and  those  of  Art.  3  are  most  con- 
venient. Of  these  last,  the  investigations  of  the  preceding 
article  show  that  eq.  (4)  is  the  only  one  which  gives  any 
conditions  of  motion  in  the  problem  under  consideration. 

Putting  the  value  of 


r 

1  = 


rdw 

(j 


Art.  7.]  TORSIONAL    OSCILLATIONS.  887 

in  eq.  (4)  of  Art.  3,  that  equation  may  take  the  form 
Gd2w 


or     ~df~-dx*=0'    •     '     •     to 

>•" 

For  brevity,  62  is  written  for  —  . 

That  dimension  of  the  cross-section  of  the  body  which 
lies  in  the  direction  of  the  radius  will  be  assumed  so  small 
that  w  may  be  considered  a  function  of  x  and  t  only.  The 
results  will  then  apply  to  small  solid  cylinders  and  all  hollow 
ones  with  thin  walls. 

The  general  integral  of  eq.  (i),  on  the  assumption  just 
made,  is  (Books'  "  Differential  Equations,"  Chap.  XV, 
Ex.  i) 


in  which  /  and  F  signify  any  arbitrary  functions  whatever. 
Now  it  is  evident  that  all  oscillations  are  of  a  periodic  char- 
acter, i.e.,  at  the  end  of  certain  equal  intervals  of  time,  w 
will  have  the  same  value.  Hence  since  /  and  F  are  arbitrary 
forms,  and  since  circular  functions  are  periodic,  there  may 
be  written 

w  --=  A  n  {  sin  (anx  +  a  ,bt)  +  sin  (anx-a  „  bt)  } 

-J5M{cos  (anx  +  aHbt)-cos  (anx-aubt)},      (2) 


in  which  an,  An,  and  Bn  are  coefficients  to  be  determined. 

Substituting  for  the  sines  and  cosines  of  sums  and  differ- 
ences of  angles, 

w  =  2  sin  anx(AH  cos  anbt  +  Btt  sin  anbt).    .     .     (3) 

Let  the  origin  of  co-ordinates  be  taken  at  the  fixed  end 
of  the  piece,  w  must  then  be  equal  to  zero,  as  is  shown  by 


888  CIRCULAR  CYLINDERS.  [Ch.  II. 

eq.  (3).  But  there  may  be  other  points  at  which  w  is  always 
equal  to  zero,  whatever  value  the  time  /  may  have.  These 
points,  called  nodes,  are  found  by  putting  w  =  o,  or 

sin  ax  =  o (4) 

This  equation  is  satisfied  by  taking 

TT      2?r      3?r  nn 

a"  =  a'     "a"'     "a"'  •"  '.  a' 

and  x  =  a ;  in  which  a  is  the  length  of  the  piece. 
Hence  at  the  distances 


a    a 
?  3' 


from  the  fixed  end  of  the  piece,  there,  will  exist  sections  which 
are  never  distorted  or  moved  from  their  positions  of  rest.  These 
are  called  nodes,  and  one  is  assumed  at  the  free  end,  although 
such  an  assumption  is  not  necessary,  since  a  is  really  the 
distance  from  the  fixed  end  to  the  farthest  node  and  not 
necessarily  to  the  free  end. 

If,  as  is  permissible,  An  and  Bn  be  written  for  twice 
those  quantities,  the  general  value  of  w  now  becomes 


Tix  f  .          nbt  nbt\ 

=  sin  —  (  Al  cos  —  +  Bl  sin  —  I 

7tX/ 

. 
a  \ 


w  = 

27ibt 


„  27rbt\ 

+  sin  --  .4,  cos  -    -  +  By  sin  - 

a  / 


+  sm  i —      ,  cos 


—  +  5,  sin  ^-} 

a  a  I 


nr.x/  .  nnbt  .    m:bt\ 

+  sin  —  (An  cos  -^-  +  Bn  sin  — j.   .     .     (5) 


Art.  7.]  TORSION  A  L    OSCILLATIONS.  889 

The  coefficients  A  and  B  are  to  be  determined  by  the 
ordinary  procedure  for  such  cases.     Let 


be  the  expression  for  the  initial  or  known  strain  at  any  point, 
for  which  the  time  t  is  zero.  Then  if  An  is  any  one  of  the 
coefficients  A, 

r\         I     &  -t/l  TT'Y" 

An=~  /  <£(*)sm--d#.    "v;V  .     .     (6) 

LI*/  0-  U 

The  velocity  at  any  point,  or  at  any  time,  will  be  given 
by 

dw  .    nxf  .          nbt      _          nbt\7ib 

-r  =  —  sm  —  (Asm  --  B.  cos  —  )  —  .  .     .     (7) 

dt  a  \    1         a  a  /  a 

In  the  initial  condition,  when  the  time  is  zero,  or  J=o, 
it  has  the  given,  or  known,  value 


Then,  as  before, 


Thus  the  most  general  value  of  w  is  completely  deter- 
mined. 

The  intensity  of  shear  at  any  place  or  time  is  given  by 

dw 
w  being  taken  from  eq.  (5). 


890  CIRCULAR  CYLINDERS.  [Ch.  II. 

The  second  case  to  be  treated  is  that  of  the  torsion  pen- 
dulum, in  which  the  mass  of  the  twisted  body  is  so  incon- 
siderable in  comparison  with  that  of  the  heavy  body,  or 
bob,  attached  to  its  free  end  that  it  may  be  neglected. 

Let  M  represent  the  mass  of  the  pendulum  bob,  and  k 
its  radius  of  gyration  in  reference  to  the  axis  about  which  it  is 
to  vibrate,  then  will  Mk~  be  its  moment  of  inertia  about  the 
same  axis. 

The  unbalanced  moment  of  torsion,  with  the  angle  of 
torsion  a,  is,  by  eq.  (9)  of  Art.  6, 


The  elementary  quantity  of  work  performed  by  this 
unbalanced  couple,  if  ft  is  the  general  expression  for  the 
angular  velocity  of  the  vibrating  body,  is 

GaIP.pdt. 

This  quantity  of  energy  is  equal  in  amount  but  opposite 
in  sign  to  the  indefinitely  small  variation  of  actual  energy 
in  the  bob  ;  hence 


Galf3dt  =-d 


But  if  a  is  the  length  of  the  piece  twisted, 
0    d(aa) 

=  and 


Art.  7.]  TORSIONAL   OSCILLATIONS.  .   891 

Multiplying  this  equation  by  2d(aa),  and  for  brevity 
putting 


then  integrating  and  dropping  the  common  factor  a2, 


When  a=av  the  value  of  the  angle  of  torsion  at  the 
extremity  of  an  oscillation,  the  bob  will  come  to  rest  and 

-  will  be  zero.     Hence 
ai 


and 


da  H    , 

^\  —  .at; 


.'.     $m-i~  =  t\~  +  (C'=o).       .     .     .     (9) 

C'  =o  because  a  and  t  can  be  put  equal  to  zero  together. 
At  the  opposite  extremities  of  a  complete  oscillation  a 
will  have  the  values 


and     (-aj. 
Putting  these  values  in  the  expression 


892  TORSION  PENDULUM.  [Ch.  II. 

and  taking  the  difference  between  the  results  thus  obtained, 
the  following  interval  of  time  for  a  complete  oscillation  will 
be  found: 

(: 

The  time  required  for  an  oscillation  is  thus  seen  to  vary 
directly  as  the  square  root  of  the  moment  of  inertia  of  the  bob 
and  the  length  of  the  piece,  and  inversely  as  the  square  root  of 
the  coefficient  of  elasticity  for  shearing  and  the  polar  moment 
of  inertia  of  the  normal  section  of  the  piece  twisted. 

The  number  of  complete  oscillations  per  second  is  -.    If 

this  number  is  the  observed  quantity,  the  following  equa- 
tion will  give  G : 


if 

The  formulas  for  this  case  should  only  be  used  when  the 
mass  of  the  cylindrical  piece  twisted  is  exceedingly  small  in 
comparison  with  M. 

Art.  8.— Thick,  Hollow  Spheres. 

In  order  to  investigate  the  conditions  of  equilibrium  of 
stress  at  any  point  within  the  material  which  forms  a  thick 
hollow  sphere,  it  will  be  most  convenient  to  use  the  equa- 
tions of  Art.  4.  As  in  the  case  of  a  thick  hollow  cylinder, 
the  interior  and  exterior  surfaces  of  the  sphere  are  supposed 
to  be  subjected  to  fluid  pressure. 

Let  r'  and  rl  be  the  interior  and  exterior  radii,  respec- 
tively. 

Let  —  p  and  —  p^  be  the  interior  and  exterior  intensities, 
respectively. 


Art.  8.]  THICK,  HOLLOW  SPHERES.  893 

Since  each  surface  is  subj  ected  to  normal  pressure  of  uni- 
form intensity  no  tangential  internal  stress  can  exist,  but 
normal  stresses  in  three  rectangular  co-ordinate  directions 
may  and  do  exist.  Consequently,  in  the  notation  of  Art.  4, 

TV  =  T<j,r  =  TW  =  o. 

With  a  given  value  of  r,  also,  a  uniform  state  of  stress 
will  exist.  Neither  N</,  nor  N^  can,  then,  vary  with  <p  or  <f>. 
By  the  aid  of  these  considerations,  and  after  omitting  R0, 
<P0,  ^o,  and  the  second  members,  the  eqs.  (i),  (2),  and  (3) 
of  Art.  4  reduce  to 


dN, 

'  r  °  ..... 

(2) 


dr  ' 


By  eq.  (2) 

Eq.  (i)  then  becomes 


Nr-N+ 


On  account  of  the  existing  condition  of  stress  which  has 
just  been  indicated  it  at  once  results  that 

ij  =  a>=o, 

and  that  p  is  a  function  of  r  only. 

Eqs.  (4)  to  (10)  of  Art.  4  then  reduce  to 


(4) 


894  THICK,  HOLLOW  SPHERES.  [Ch.  IT. 

After  substitution  of  these  quantities,  eq.  (3)  becomes 
l(Pp      2rdp—2pdr\        ^d2o  dp  p 

•FT:  -f 5-H +  2G--7-.r  +  4,6 — ; 4U^  =0, 


One  integration  gives 

dp     20 


(7) 


Hence  0,  the  rate  of  variation  of  volume,  is  a  constant 
quantity.  Eq.  (7)  may  take  the  form 

rdf}  +  2f)dr  =  cr  dr. 

As  it  stands,  this  equation  is  not  integrable,  but,  by  in- 
specting its  form,  it  is  seen  that  r  is  an  integrating  factor. 
Multiplying  both  sides  of  the  equation,  then,  by  r, 

2rpdr  =d(r2p)  =cr7dr; 

...     (8) 

Substituting  from  eqs.  (7)  and  (8)  in  eq.  (5), 


A     4bG 
A-     ,. 


It  is  obvious  what  .4  represents. 


Art.  8,];  THICK,  HOLLOW  SPHERES.  895 

When  /  and  rt  are  put  for  r,  Nr  becomes  -p  and  -ft. 
Hence 


and 


These  equations  express  the  conditions  involved  in  eqs. 

(13),  (14),  and  (15)  of  Art.  2. 
The  last  equations  give 


r/3_    3  • 


These  quantities  make  it  possible  to  express  Nr  and  N+ 
independently  of  the  constants  of  integration,  c  and  b,  for 
those  intensities  become 


ft-pr         Pl-prr      i. 

/S.S  ^S.S  r3, 


Thus  it  is  seen  that  A^=A^  has  its  greatest  value  for 
the  interior  surface;  that  intensity  will  be  called  h. 

It  is  now  required  to  find  rl  —  rf=tin  terms  of  h,  p,  and  pv 
If  r  =r'  in  eq.  (n)f 


896  THICK,  HOLLOW  SPHERES.  [Ch.  II. 

Dividing  this  equation  by  r'3  and  solving, 
rta 


If  the  intensities  p  and  pl  are  given  for  any  case,  eq.  (12) 
will  give  such  a  thickness  that  the  greatest  tension  h  (sup- 
posing pl  considerably  less  than  p)  shall  not  exceed  any 
assigned  value.  If  the  external  pressure  is  very  small  com- 
pared with  the  internal,  p1  may  be  omitted. 

The  values  of  A  and  46*6  allow  the  expressions  for  c  and 
b  to  be  at  once  written. 

If  p^  is  greater  than  p,  nothing  is  changed  except  that 
N+—N+  becomes  negative,  or  compression. 


CHAPTER  III. 
THEORY  OF   FLEXURE. 
Art.  9.  —  General  Formulae. 

IF  a  prismatic  portion  of  material  is  either  supported  at 
both  ends,  or  fixed  at  one  or  both  ends,  and  subjected  to 
the  action  of  external  forces  whose  directions  are  normal 
to,  and  cut,  the  axis  of  the  prismatic  piece,  that  piece  is  said 
to  be  subjected  to  "  flexure."  If  these  external  forces  have 
lines  of  action  which  are  oblique  to  the  axis  of  the  piece,  it 
is  subjected  to  combined  flexure  and  direct  stress. 

Again,  if  the  piece  of  material  is  acted  upon  by  a  couple 
having  the  same  axis  with  itself,  it  will  be  subjected  to  "  tor- 
sion." 

The  most  general  case  possible  is  that  which  combines 
these  three,  and  some  general  equations  relating  to  it  will 
first  be  established. 

The  co-ordinates  axis  of  X  will  be  taken  to  coincide  with 
the  axis  of  the  prism,  and  it  will  be  assumed  that  all  external 
forces  act  upon  its  ends  only.  Since  no  external  forces  act 
upon  its  lateral  surface,  there  will  be  taken 


retaining  the  notation  of  Art.  2.  These,  conditions  are  not 
strictly  true  for  the  general  case,  but  the  errors  are,  at  most, 
excessively  small  for  the  cases  of  direct  stress  or  flexure,  or 

897 


898  THEORY  OF  FLEXURE.  [Ch.  III. 

for  a  combination  of  the  two.     By  the  use  of  eqs.  (12),  (21), 
and  (22)  of  Art.  i  the  conditions  just  given  become 

r      (du    dv      (h^\J_d^_  /  \ 

+  ^~ 


r      (du     dv      dw        dw 


dv     dw 


Eqs.  (i)  and  (2)  then  give 
dv      dw 


(4) 


In  consequence  of  eq.  (4)  eqs.  (i)  and  (2)  give 

dv     dw          du 


By  the  aid  of  eq.  (5)  and  the  use  of  eqs.  (n),  (13),  and 
(20)  of  Art.  i,  in  eqs.  (10),  (n),  and  (12)  of  Art.  2  (in  this 
case  X0  =  yo=Z0  =  o),  there  will  result 


d2u      d2v 
^Ty  +  d^ 

d2u      d2w 


doc  dz^  doc 


,  . 
(8) 


Eqs.  (3),  (5),  (6),  (7),  and  (8)  are  five  equations  of 
condition  by  which  the  strains  u,  v,  and  .w  are  to  be  deter- 
mined. 


Art.  9.]  GENERAL  FORMUL/E. 

Let  eq.  (6)  be  differentiated  in  respect  to  x: 

d*u          dzu 


dx3     dy2  dx     dzrdx=' 

From  this  equation  let  there  be  subtracted  the  sum  of 
the  results  obtained  by  differentiating  eq.  (7)  in  respect  to  y 
and  (8)  in  respect  to  z: 

dsu         dsv          d5w 


In   this   equation   substitute   the   results   obtained   by 
differentiating  eq.  (5)  twice  in  respect  to  x,  there  will  result 


d* 


This  result,  in  the  equation  immediately  preceding  eq. 
(9),  by  the  aid  of  eq.  (5)  will  give 


dx2  dy 


=  o. 


After  differentiating  eq.  (7)  in  respect  to  y,  and  substi- 
tuting the  value  immediately  above, 


d'u 


Eqs.  (9)  and  (10)  enable  the  second  equation  preceding 
eq.  (9)  to  give 


d^\ 

dx) 


dz 


goo  THEORY  OF  FLEXURE.  [Ch.  III. 

Let  the  results  obtained  by  differentiating  eq.  (7)  in 
respect  to  z  and  (8)  in  respect  to  y  be  added : 

d3u  d3v    .     d3w 

dx  dy  dz     dx2  dz     dx2  dy 

The  sum  of  the  second  and  third  terms  of  the  first  mem- 
ber of  this  equation  is  zero,  as  is  shown  by  twice  differentiat- 
ing eq.  (3)  in  respect  to  x.  Hence 

Jdu 
d*u  (dx,      _^ 


dy  dz  dx      dy  dz 

Eqs.  (9),    (10),    (n),    and    (12)    are   sufficient  for  the 

determination  of  the  form  of  the  function  -y-,  if  it  be  assumed 

ax 

to  be  algebraic,  for 

Eq.     (9)  shows  that  x2  does  not  appear  in  it ; 
"     (10)      " 


"     (12)  yz 

The  products  xz  and  xy  may,  however,  be  found  in  the 
function.  Hence  if  a,  alf  a2,  b,  bv  and  b2  are  constants, 
there  may  be  written 

du  ,     v 

•     •     •     (i3) 


"Eq.  (5)  then  gives 

.    .     (14) 


Art.  9.]  GENERAL  FORMUL/E.  901 

Substituting  from  eq.  (13)  in  eqs.  (7)  and  (8), 


(15) 


The  method  of  treatment  of  the  various  partial  deriva- 
tives in  the  search  for  eqs.  (13)  and  (14)  is  identical  with  that 
given  by  Clebsch  in  his  "  Theorie  der  Elasticitat  Fester 
K  or  per." 

It  is  to  be  noticed  that  the  preceding  treatment  has  been 
entirely  independent  of  the  form  of  cross-section  or  direction 
of  external  forces. 

It  is  evident  from  eqs.  (13)  and  (14)  that  the  constant  a 
depends  upon  that  component  of  the  external  force  which 
acts  parallel  to  the  axis  of  the  piece  and  produces  tension  or 
compression  only.  For  (pages  9,  10)  it  is  known  that 
if  a  piece  of  material  be  subjected  to  direct  stress  only, 

du  dv    dw 

-j-  =  a     and    -r=-r  =  —w, 

dx  dy     dz 

the  negative  sign  showing  that  ra  is  opposite  in  kind  to  a, 
both  being  constant. 

Again,  if  z  and  y  are  each  equal  to  zero,  eq.  (13)  shows 
that 

du          , 
-j-  =  a  +  ox. 
dx 

Hence  bx  is  a  part  of  the  rate  of  strain  in  the  direction  of  x 
which  is  uniform  over  the  whole  of  any  normal  section  of  the 
piece  of  material,  and  it  varies  directly  with  x.  But  such  a 


902  THEORY  OF  FLEXURE.  [Ch.  Ilf. 

portion  of  the  rate  of  strain  can  only  be  produced  by  an 
external  force  acting  parallel  to  the  axis  of  X,  and  whose 
intensity  varies  directly  as  x.  But  in  the  present  case 
such  a  force  does  not  exist.  Hence  b  must  equal  zero. 

The  eqs.  (13),  (14),  (15),  and  (16)  show  that  av  bl  and 
dv  b2  are  symmetrical,  so  to  speak,  in  reference  to  the  co- 
ordinates z  and  y,  while  eqs.  (13)  and  (14)  show  that  the  nor- 
mal intensity  A^  is  dependent  on  those,  and  no  other,  ..con- 
stants in  pure  flexure  in  which  a  =  o.  It  follows,  there- 
fore, that  those  two  pairs  of  constants  belong  to  the  two 
cases  of  flexure  about  the  two  axes  of  Z  and  Y. 

No  direct  stress  A/^  can  exist  in  torsion,  which  is  simply  a 
twisting  or  turning  about  the  axis  of  X. 

Since  the  generality  of  the  deductions  will  be  in  no  man- 
ner affected,  pure  flexure  about  the  axis  of  Y  will  be  con- 
sidered. For  this  case 


Making  these  changes  in  (13)  and  (14), 
du 


dv  _dw          du 

dy    dz  ~     ~r~cfoc=  ~r(aiz  +  b1xz).  .     .     .     (18) 

*  _du     dv     dw 

dx     dv     dz  ~z(ai~T~t>ix)(I  ~  2rh  •     -     (19) 


Also, 


(20) 


Art.  9-1  !  GENERAL  FORMULAE.  903 

since 

2G(r+i)=E. 

Taking  the  first  derivative  of  N19 

V).      •  ;  .....     (21) 


This  important  equation  gives  the  law  of  variation  of 
the  intensity  of  stress  acting  parallel  to  the  axis  of  a  bent 
beam,  in  the  case  of  pure  flexure  produced  by  forces  exerted 
at  its  extremity.  That  equation  proves  that  in  a  given  nor- 
mal section  of  the  beam,  .whatever  may  be  the  form  of  the 
section,  the  rate  of  variation  of  .the  normal  intensity  of  stress  is 
constant  ;  the  rate  being  taken  along  the  direction  of  the  external 
forces. 

It  follows  from  this  that  N^  must  vary  directly  as  the 
distance  from  some  particular  line  in  the  normal  section 
considered  in  which  its  value  is  zero.  Since  the  external 
forces  F  are  normal  to  the  axis  of  the  beam  and  direction 
of  Nv  and  because  it  is  necessary  for  equilibrium  that  the 
sum  of  all  the  forces  N1dy  dz,  for  a  given  section,  must  be 
equal  to  zero,  it  follows  that  on  one  side  of  this  line  tension 
must  exist,  and  on  the  other  compression. 

Let  A/"  represent  the  normal  intensity  of  stress  at  the 
distance  unity  from  the  line,  b.  the  variable  width  of  the 
section  parallel  to  y,  and  let  A  =  bdz.  The  sum  of  all  the 
tensile  stress  in  the  section  will  be 

tz'  fz' 

Nz4=N\  zA. 

Jo  Jo 

The  total  compressive  stress  will  be 

.'.I...'.'          .  -     ,.  fO 

N\     zd. 

J   -2! 


904 


THEORY  OF  FLEXURE. 


[Ch.  III. 


The  integrals  are  taken  between  the  limits  o  and  the  greatest 
value  of  z  in  each  direction,  so  as  to  extend  over  the  entire 
section.  In  order  that  equilibrium  may  exist,  therefore, 


0—4- 


FlG.    I. 


=  0. 


(22) 


Eq.  (22)  shows  that  the  line  of  no  stress  must  pass  through 
the  centre  of  gravity  of  the  normal  section. 

This  line  of  no  stress  is  called  the  neutral  axis  of  the 
section.  Regarding  the  whole  beam,  there  will  be  a  sur- 
face which  will  contain  all  the  neutral  axes  of  the  different 
sections,  and  it  is  called  the  neutral  surface  of  the  bent 
beam.  The  neutral  axis  of  any  section,  therefore,  is  the 
line  of  intersection  of  the  plane  of  section  and  neutral  sur- 
face. 

Hereafter  the  axis  of  X  will  be  so  taken  as  to  traverse 
the  centres  of  gravity  of  the  different  normal  sections 
before  flexure.  The  origin  of  co-ordinates  will  then  be 


Art,  9  1 


GENERAL  FORMULA. 


905 


FIG.  2. 


taken  at  the  centre  of  gravity  of  the  fixed  end  of  the  beam, 
as  shown  in  Fig.  i. 

The  value  of  the  expression  (al  +  blx),  in  terms  of  the 
external  bending  moment,  is  yet  to  be  determined.  Con- 
sider any  normal  section  of  the  beam  located 
at  the  distance  x  from  0,  Fig.  i,  and  let 
OA  =/.  Also,  let  Fig.  2  represent  the  sec- 
tion considered,  in  which  BC  is  the  neutral 
axis  and  df  and  d1  the  distances  of  the  most 
remote  fibres  from  BC.  Let  moments  of  all 
the  forces  acting  upon  the  portion  (l  —  x)  of 
the  beam  be  taken  about  the  neutral  axis  BC.  .If,  again,  b 
is  the  variable  width  of  the  beam,  the  internal  resisting 
moment  will  be 

[      Nlbzdz=E(al  +  blx)\     z\bdz. 

J  -di  J  -di 

But  the  integral  expression  in  this  equation  is  the  moment 
of  inertia  of  the  normal  section  about  the  neutral  axis,  which 
will  hereafter  be  represented  by  /.  The  moment  of  the 
external  force,  or  forces,  F,  will  be  F(l  —  x),  and  it  will  be 
equal,  but  opposite  in  sign,  to  the  internal  resisting  moment. 
Hence 


(23) 


(24) 


Substituting  this  quantity  in  eq.  (16), 


M 


dx 


(25) 


9o6  THEORY  OF  FLEXURE.  [Ch.  III. 

It  has  already  been  seen  (page  38)  that  eq.  (25)  is  one 
of  the  most  important  equations  in  the  whole  subject  of  the 
"Resistance  of  Materials." 

An  equation  exactly  similar  to  (25).  may  of  course  be 
written  from  eq.  (15);  but  in  such  an  expression  M  will 
represent  the  external  bending  moment  about  an  axis  par- 
allel to  the  axis  of  Z. 

No  attempt  has  hitherto  been  made  to  determine  the 
complete  values  of  u,  v,  and  w,  for  the  mathematical  opera- 
tions involved  are  very  extended.  If,  however,  a  beam  be 
considered  whose  width,  parallel  to  the  axis  of  Y,  is  indefi- 
nitely small,  u  and  w  may  be  determined  without  difficulty. 
The  conclusions  reached  in  this  manner  will  be  applicable 
to  any  long  rectangular  beam  without  essential  error. 

If  y  is  indefinitely  small,  all  terms  involving  it  as  a  factor 
will  disappear  in  u  and  w ;  or,  the  expressions  for  the  strains  u 
and  w  will  be  junctions  of  z  and  x  only.  But  making  u  and  w 
functions  of  z  and  x  only  is  equivalent  to  a  restriction  of 
lateral  strains  to  the  direction  of  z  only,  or  to  the  reduction 
of  the  direct  strains  one  half,  since  direct  strains  and  lateral 
strains  in  two  directions  accompany  each  other  in  the  un- 
restricted case.  Now  as  the  lateral  strain  in  one  direction 
is  supposed  to  retain  the  same  amount  as  before,  while  the 
direct  strain  is  considered  only  half  as  great,  the  value  of 
their  ratio  for  the  present  case  will  be  twice  as  great  as  that 
used  on  pages  9  to  12.  Hence  2r  must  be  written  for  r,  in 
order  that  that  letter  may  represent  the  ratio  for  the  unre- 
stricted case,  and  this  will  be  done  in  the  following  equations. 

Since  w  and  u  are  independent  of  y, 

dw    du  dv 

j—  =  -j-  =  o,     and     To  =  G~r". 
dy     dy  dx 

But,  by  eq.  (14), 

v-  -  2r(o1  +  btfzy  +  f(x,  z). 


Art.  9-]  GENERAL   FORMULA.  907 

By  eq.   (3),  since 

dw 
dy=°> 

dv  d  f 

f(x,  z)  =o. 


This  equation,  however,  involves  a  contradiction,  for  it 
makes  f(x,  z)  equal  to  a  function  which  involves  yt  which  is 
impossible.  Hence 

f(x,z)  =o. 
Consequently 

dv 


which  is  indefinitely  small  compared  with 

dv 

^=-2r(a1  +  blx)z, 

and  is  to  be  considered  zero 
Because  f(x,z)=o, 

dv 

~j~  =  —  iro.zy. 

dx 

This  quantity  is  indefinitely  small;  hence 


is  of  the  same  magnitude. 

Under  the  assumption  made  in  reference  to  y,  there  may 
be  written,  from  eqs.  (17)  and  (18), 


....     (26) 
.    •     •     •     (27) 


9°8  THEORY  OF  FLEXURE.  [Ch.  III. 

Using  eq.  (26)  in  connection  with  eq.  (6), 


By  two  integrations, 

".  .  (28) 


Using  eq.  (27)  in  connection  with  eq.  (8), 


By  two  integrations, 


a.x 

-J 


The  functions  u  and  w  now  become 


.   x2       b.z3 
u—ajcz+bi—z —-c'z  +  c";  .    .    .     (29) 


w  =  _  ra^  -  rb.xz2  -  b       -  °-  +  c,x  +  cir    .     (30) 


The  constants  of  integration  cf,  c"  ,  etc.,  depend  upon 
the  values  of  u  and  w,  and  their  derivatives,  for  certain 
reference  values  of  the  co-ordinates  x  and  z,  and  also 
upon  the  manner  of  application  of  the  external  forces,  F,  at 
the  end  of  the  beam,  Fig.  i  .  The  last  condition  is  involved 
in  the  application  of  eqs.  (13),  (14),  and  (15)  of  Art.  2. 


Art.  9.]  GENERAL  FORMUL/E. 


909 


In  Fig.  i  let  the  beam  be  fixed  at  0.     There  will  then 
result,  for  x  =  6  and  2  =  0, 

du 

In  virtue  of  the  last  condition, 
c"=cn=o. 

In  consequence  of  the  first, 

c'=o. 

After  inserting  these  values  in  eqs.  (29)  and  (30), 
du  ,  x2 


dw  .   x 

—  .  -rblz*-b1- 


The  surface  of  the  end  of  the  beam,  on  which  F  is  applied, 
is  at  the  distance  /  from  the  origin  0  and  parallel  to  the 
plane  ZY.  Also,  the  force  F  has  a  direction  parallel  to  the 
axis  of  Z.  Using  the  notation  of  eqs.  (13),  (14),  and  (15)  of 
Art.  2,  these  conditions  give 


=  o,     cosr=o, 

COS7T=0,       COS  7  =  0,       COS|0  =  I. 


THEORY  OF  FLEXURE.  [Ch.  III. 

Since,  for  x  =  lt 


eqs.  (24)  and  (20)  give  N1=o  for  all  points  of  the  end  sur- 
face.     Eq.  (15)  is,  then,  the  only  one  of  those  equations 
which  is  available  for  the  determination  of  cv 
That  equation  becomes  simply 

T,-P. 

For  a  given  value  .of  z,  therefore,  any  value  may  be  as- 
sumed f  :>r  T2.  For  the  upper  and  lower  surfaces  of  the  beam 
let  the  intensity  of  shear  be  zero;  or  for  z=  ±d  let  T2  =  o. 
Hence,  by  eq.  (31), 


Fh 
-V-*«).  ......     (33) 


The  constants  at  and  bl  still  remain  to  be  found.  The 
only  forces  acting  upon  the  portion  (/  —  x)  of  the  beam  are 
F  and  the  sum  of  all  the  shears  T2  which  act  in  the  section  x. 
Let  Ay  be  the  indefinitely  small  width  of  the  beam,  which, 
since  z  is  finite,  is  thus  really  made  constant.  The  princi- 
ples of  equilibrium  require  that 

f+  T2.4y.dz  =  Gb1(i+r)r  (d\  Ay.dz-z\  Ay  .dz]  =F. 

J  —d     '  J  —d 

The  first  part  of  the  integral  will  be  2  dyds,  and  the  second 
part  will  be  the  moment  of  inertia  of  the  cross-section  (made 


Art.  9.]  GENERAL  FORMUL/E.  gji 

rectangular  by  taking  Ay  constant)  about  the  neutral  axis. 
Hence 

i+r)7-F,     or    &i--.    •     (33) 


•.-.T,  -((*'-*»).     .....     (34) 

If  x=o  in  eq.  (24), 

FJ 


(35) 


Thus  the  two  conditions  of  equilibrium  are  involved  in 
the  determination  of  av  and  bv  The  complete  values  of  the 
strains  u  and  w  are,  finally, 


(37) 


These  results  are  strictly  true  for  rectangular  beams  of 
indefinitely  small  width,  but  they  may  be  applied  to  any 
rectangular  beam  fixed  at  one  end  and  loaded  at  the  other, 
with  sufficient  accuracy  for  the  ordinary  purposes  of  the 
civil  engineer.  It  is  to  be  remembered  that  the  load  at  the 
end  is  supposed  to  be  applied  according  to  the  law  given 
by  eq.  (34),  a  condition  which  is  never  realized.  Hence 
these  formulae  are  better  applicable  to  long  than  short 
beams. 


912  THEORY  OF  FLEXURE.  [Ch.  III. 

The  greatest  value  of  T2,  in  eq.   (34),  is  found  at  the 
neutral  axis  by  making  2=0;  for  which  it  becomes 


F 

—7  is  the  mean  intensity  of  shear  in  the  cross-section; 

hence  the  greatest  intensity  of  shear  is  once  and  a  half  as 
great  as  the  mean. 

In  eq.  (36),  if  2  =  0,  u  =  o.  Hence  no  point  of  the  neu- 
tral surface  suffers  longitudinal  displacement. 

In  eq.  (37)  the  last  term  of  the  second  member  is  that 
part  of  the  vertical  deflection  due  to  the  shear  at  the  neu- 
tral surface,  as  is  shown  by  eq.  (38).  The  first  term  of 
the  second  member,  being  independent  of  x,  is  that  part 
of  the  deflection  which  arises  wholly  from  the  deformation 
of  the  normal  cross-section. 

The  usual  modification  of  this  treatment,  designed  to 
supply  formulae  for  the  ordinary  experience  of  the  engineer, 
has  already  been  given  in  preceding  articles. 


APPENDIX  II. 

CLAVARINO'S  FORMULA. 

IN  Art.  13  reference  is  made  to  .  Clavarino's  formula 
for  thick  cylinders.  It  will  be  sufficient  here  to  establish 
the  equation  for  the  circumferential  or  hoop  tension  in 
a  thick  cylinder  to  illustrate  Clavarino's  fundamental  idea. 

If  /  represents  the  unit  strain  in  the  direction  of.  a 
tensile  force  acting  alone  and  whose  intensity  is  T,  and 
if  I'  is  the  unit  longitudinal  strain  in  the  same  direction 
under  the  same  stress  T  but  with  two  intensities  of  com- 
pressive  stress  R  and  5  acting  at  right  angles  to  each  other 
and  to  the  stress  T  with  corresponding  direct  unit  strains  l\ 
and  /2,  and  finally  if  r  is  the  ratio  of  the  lateral  strain 
divided  by  the  direct  or  longitudinal  strain,  then  will 


(i 


According  to  Clavarino's  view  a  lateral  strain  repre- 
sents the  action  or  an  actual  force  or  stress  with  an  in- 
tensity equal  to  the  modulus  of  elasticity  E  multiplied 
by  the  lateral  unit  strain.  Consequently  he  considered 


In  the  case  of  the  thick  cylinder  T  is  the  intensity  of 
stress  originally  established  by  Lame  and  given  by  eq. 
(16)  Art.  5  of  Appendix  I,  while  R  is  the  radial  compres- 
sion given  by  eq.  (15)  of  the  same  Art.,  and  5  is  the  intensity 

913 


914  CLAVARmorS  FORMULA.  App.  II. 

of  longitudinal  tensile  stress  existing  if  the  cylinder  has 
closed  ends  and  it  is  found  by  eq.  (3) ; 

*-*£££ (3) 

As  5  is  a  tensile  stress  and  causes  a  negative  lateral 
strain  the  term  rS  in  eq.  (2)  must  have  the  negative  sign. 
Again,  eq.  (15),  Art.  5,  of  Appendix  I  is  so  written  as  to 
make  R  negative.  Hence,  for  the  present  purpose,  eq.  (2) 
must  be  written : 

Substituting  the  values  of  R  and  T  from  eqs.  (15)  and 
(16),  Art.  5,  Appendix  I,  and  the  value  of  5  from  eq.  (3), 
in  eq.  (4)  and  taking  r  =  \, 


If  r  -r'  in  eq.  (5),  the  greatest  value  of  Tr  becomes: 
Finally,  if  £1=0, 


If  the  stress  S=o  the  corresponding  modifications  of 
the  formulae  are  obvious. 

Eq.  (6)  gives  for  the  exterior  radius; 


App.  II.]  CLAVARINO'S  FORMULA.  915 

These  equations  illustrate  Clavarino's  formulae.  For 
the  reasons  given  fully  in  Art.  13,  they  can  be  considered 
approximate  only. 

Related  closely  to  Clavarino's  method  is  that  procedure 

E> 

of  arbitrarily  assuming  T-\ —  =  constant  in  an  analysis  of 

o 

the  stresses  in  the  wall  of  a  thick  cylinder.     At  best  the 
results  are  but  approximate. 


APPENDIX   III. 

RESISTING  CAPACITY  OF  NATURAL   AND 
ARTIFICIAL  ICE. 

In  the  early  part  of  1913  two  graduating  students  in 
Civil  Engineering,  Messrs.  A.  F.  Lipari  and  R.  M.  Marx, 
at  Columbia  University,  acting  under  the  immediate  direc-' 
tion  of  Mr.  J.  S.  Macgregor,  in  charge  of  the  testing  labora- 
tory of  the  Department  of  Civil  Engineering,  conducted  a 
series  of  physical  tests  of  natural  and  artificial  ice,  both  in 
compression  and  in  flexure.  These  tests  were  made  with 
scrupulous  care  as  to  the  application  of  loads  to  test  pieces 
and  in  the  quantitative  determination  of  results.  The 
test  pieces  in  compression  were  subjected  to  their  loads  in 
the  cooling  apparatus  employed.  The  compression  tests  of 
the  natural  ice  were  made  with  the  load  applied  in  some 
cases  normal  to  its  natural  surface  and  in  other  tests  parallel 
to  that  surface,  in  other  words  normal  to  its  bed  and  parallel 
to  its  bed. 

The  behavior  of  the  two  kinds  of  ice  in  the  tests  was 
quite  different  in  some  respects.  A  block  of  clear  artificial 
ice  would  soon  be  clouded  under  a  gradual  application  of 
loading  by  the  formation  of  crystals,  which  finally  would 
determine  the  lines  of  compressive  failure;  while  the  ten- 
dency of  the  natural  ice  was  to  separate  and  fail  in  columns. 
In  both  cases,  however,  there  was  a  distinct  tendency  to 
shear  on  oblique  planes,  making  an  angle  of  about  45°  with 
the  direction  of  loading.  The  separation  along  these  shear 
planes  was  distinctly  marked  in  many  specimens. 

916 


App.  III.] 


RESISTING  CAPACITY  OF  ICE. 


917 


In  general  the  height  of  the  compression  test  specimens 
was  about  twice  the  greatest  cross  dimensions,  but  the  larg- 
est specimens  tested  were  exceptions  to  this  observation. 
The  accompanying  table  gives  a  concise  statement  of  the 
results  of  the  fifty-seven  tests  of  natural  ice  in  compression 
and  of  the  thirty-one  compressive  tests  of  the  artificial  ice. 


TABLE  I. 

NATURAL   ICE   IN   COMPRESSION. 


Size  of 

Test  Pieces. 

Number  of 
Tests. 

Ult.  Comp.  Resistances 
Pounds  per  Sq.  In. 

Max. 

Mean. 

Min. 

3.25  ins 
9.8    ins 

by    3.75  ins. 
to 
by  13.9    ins. 

57 

1132 

543 

100 

ARTIFICIAL   ICE    IN   COMPRESSION 


3       ins.  by    3.2    ins. 

to 
10.5    ins.  by  10.2    ins. 


368 


185 


The  dimensions  of  the  cross-sections  of  the  test  pieces 
are  seen  to  vary  greatly.  The  number  of  pieces  tested  with 
the  larger  cross-sections  was  not  enough  to  establish  any 
definite  relation  between  the  ultimate  compressive  resist- 
ances per  square  inch  and  the  areas  of  the  cross-sections  of 
the  test  pieces.  Within  the  limits  of  these  tests  there  ap- 
pears to  be  little,  if  any,  material  variation  of  ultimate 
resistance  with  the  increase  of  cross-section. 

It  is  important  to  observe  that  the  ultimate  resistance 
of  the  artificial  ice  is  much  less  than  that  of  the  natural. 
In  fact,  the  mean  ultimate  resistance  of  the  natural  ice  is 
nearly  three  times  as  great  as  the  mean  ultimate  resistance 


pi8  RESISTING  CAPACITY  OF  ICE.  [App.  III. 

of  the  artificial,  and  about  the  same  relation  holds  for  the 
maximum  intensities. 

The  temperature  of  the  test  pieces  as  determined  by 
thermo-couples  during  the  actual  procedure  of  testing  ranged 
generally  from  about  +28°  Fahr.  to  about  freezing.  It  is 
probable  that  the -temperature  of  the  ice  was  considerably 
lower  than  indicated  by  the  apparatus. 

The  test  pieces  were  not  selected  with  any  special  care, 
but  were  fair  averages  of  natural  and  artificial  ice  as  or- 
dinarily sold  in  quantities  for  the  usual  purpose  of  city 
consumption.  Naturally  the  quality  varied  materially  in 
many  blocks  as  bought,  causing  correspondingly  wide 
variations  in  the  ultimate  resistances  determined.  The 
results  of  these  compressive  tests  show  that  sound  natural 
ice, at  about  the  temperatures  indicated  may  be  expected 
to  give  on  the  average  an  ultimate  resistance  of  about 
500  Ibs.  per  sq.  in.,  with  a  range  of  perhaps  100  to  1000 
Ibs.  per  sq.  in.  The  artificial  ice  tested  appears  to  have 
had  about  one-third  the  ultimate  resistance  only  of  the 
natural  ice. 

In  some  cases  the  test  pieces  of  natural  ice  appeared  to 
give  somewhat  greater  ultimate  resistances  when  tested 
on  their  beds  than  when  tested  on  edge.  In  scrutinizing 
the  whole  list,  however,  there  appears  to  be  but  little,  if  any, 
difference.  Hence  no  distinction  of  this  kind  has  been  made 
in  Table  I,  but  all  the  tests  have  been  treated  as  of  one 
group. 

Table  II  shows  the  results  of  testing  beams  of  both 
natural  and  artificial  ice  with  loads  applied  at  the  centre 
of  span.  The  effective  span  in  all  cases  was  18  inches. 
The  normal  cross-sections  of  the  beams  were  square  and 
varied  but  little  from  3.5  inches  by  3.5  inches.  There 
were  nine  such  tests  of  beams  of  natural  ice  and  twelve  of 
beams  of  artificial  ice.  The  modulus  of  rupture  is  the  usual 


App.  III.] 


RESISTING  CAPACITY  OF  ICE. 


919 


so-called  intensity  of  stress  in  the  extreme  fibre.  It  is 
difficult  to  state  whether  the  ice  failed  by  tension  or  com- 
pression. In  some  cases  there  was  evidence  of  partial 
failure  at  least  by  internal  shear.  Some  of  these  beams  were 
placed  so  as  to  be  loaded  on  their  beds,  so  to  speak,  and  some 
on  edge,  but  on  the  whole  there  appeared  to  be  little  dif- 
ference in  the  results.  Occasionally  there  appeared  to  be  a 
tendency  to  fail  in  such  manner  as  to  exhibit  the  "bedding" 
planes. 

TABLE  II 

BEAMS   OF   NATURAL  ICE 
LOAD  AT  CENTRE  OF  SPAN 


Span. 

Number  of  Tests. 

Modulus  of  Rupture.      Pounds  per  Sq.  In. 

Max. 

Mean. 

Min. 

1  8  ins. 

9 

351 

247 

140 

BEAMS   OF   ARTIFICIAL  ICE 


1 8  ins. 


12 


138 


122 


There  is  the  same  inferiority  of  ultimate  resistance  of 
the  artificial  ice  beams  as  in  compression,  but  the  artificial 
ice  beams  show  a  little  less  than  half  the  modulus  of  rupture 
given  by  the  natural  ice  beams. 


INDEX. 


Adhesion  between  bricks  and  stones 

and  cement  mortars,  373-375 
Adhesive  shear  or  bond,  592-598,  633 
Alloys    of    copper,    tin,    aluminum, 

zinc  in  tension,  346-362 
Alloys  of  copper,  tin,  zinc  in  torsion, 

193-195,  546 
Alloys  of  copper,  tin,  zinc  in  beams, 

355,  56i,  562 
Aluminum,  354,  355 
Aluminum  alloys  in  bending,  560,  561 
Aluminum,     alloys    of,    in    tension, 

352-358 
Aluminum,     alloys    of,    in    torsion, 

193-196 

Aluminum-zinc  beams,  354 
Angles,  steel,  as  columns,  496-500 
Annealing  of  steel,  338,  339 


B 

Balanced  economic  steel  reinforce- 
ment, 608-613,  617-618 

Batten  plates,  508 

Beams  of  ice,  918 

Beams,  solid,  rectangular,  and  cir- 
cular, 554-562 

Bearing  capacity  of  rivets,  441 

Bending  and  direct  stress  combined, 
254-267 

Bending  and  direct  stress  in  eye-bars, 
255-267 


Bending  and  torsion  combined,  246 
Bending    moments    and    shears    in 

general,  64 
Bending   moments   in   concrete-steel 

beams,  614-616,  618-619 
Brass,  349~35i 

Brick  masonry  beams,  584,  585 
Brick  piers  or  columns,  413-417 
Bricks,  adhesion  between  cement 

and,  373-375 
Bricks  and  brick  piers  in  compression, 

409-419 

Bricks  in  shearing,  550 
Bridge  portal,  stresses  in,  789 
Briquette,  Am.  Soc.  C.  E.  standard 

for  cement  tests,  372 
Bronzes  and  brass,  Board  of  Water 

Supply,  N.  Y.  City,  359,  360 
Building  stones,  420 
Bulk  modulus,  19 


Castings,  steel,  322 
Cast-iron  beams,  560 
Cast-iron  columns,  520-527 
Cast-iron,  elastic  limit,  286 
Cast-iron,  fatigue  of,  295 
Cast-iron  flanged  beams,  662-664 
Cast-iron  in  compression,  388 
Cast-iron  in  shearing,  544 
Cast-iron  in  torsion,  192-193,  544 
Cast-iron,     modulus     of     elasticity, 
286-290,  294,  389 

921 


922 


INDEX. 


Cast-iron,   remelting  and  continued 

fusion  of,  294 
Cast-iron,  tensile  resilience  of,  286- 

294 

Cast-iron,  tensile  strain  diagram,  285 
Cast-iron,     ultimate    tensile     resist- 
ance, 292,  294 
Cement  in  compression,  395 
Cement  in  tension,  362-377 
Cement  mortar  in  compression,  395 
Cement  mortar  in  tension,  362-377 
Chemical  elements  in  steel,  343 
Chrome  vanadium  steel,  329-331 
Cinder     concrete     in     compression, 

405-407 

Cinder  concrete  in  tension,  365 
Circular  cylinders,  torsion  of,  884 
Clavarino's  formula,  48,  913 
Coefficient   of    elasticity,  see    Mod- 
ulus of  elasticity. 
Collapse  of  flues,  774-778 
Column  design,  505-520 
Columns  of  cast-iron,  520-527 
Columns  of  concrete,  408 
Columns  of  timber,  528-529 
Columns,  long,  169 
Columns,    long,    wrought   iron    and 

steel,  490-520 
Combined  bending  and  compression, 

268 
Combined  bending  and  direct  stress, 

254 

Combined  bending  and  torsion,  246 
Common  theory  of  flexure,  49,  99 
Common  theory  of  flexure  for  beam 

of  two  materials,  156 
Common  theory  of  torsion,  182-196 
Composite    material,    elastic    action 

of,  749 

Compression,  385 
Compressive  resistance  of  cast-iron, 

388 
Compressive     resistance    of     steel, 

389-391 


Compressive   resistance   of   wrought 

iron,  387,  388 
Compressive  stress,  4,  385 
Concrete  columns,  408 
Concrete    columns,   reinforced,  641- 

655 

Concrete  beams,  575-583 
Concrete  in  compression,  395-409 
Concrete-steel,  adhesive  shear,  592- 

598 

Concrete-steel  beams,  600-640 
Concrete-steel     beams,     design     of, 

629-640 
Concrete-steel,  modulus  of  elasticity, 

633 

Concrete-steel  members,  588-658 
Concrete-steel    theory,    by    common 

theory  of  flexure,  591-620 
Connections,  435-473 
Connections,  pin,  470-473 
Connections,    riveted     joints,     435- 

470 

Continuous  beams  in  general,  118 
Copper,  alloys  of,  in  tension,  346- 

362 

Copper  in  compression,  396 
Copper  in  shearing  and  torsion,  546 
Copper  in  tension,  347-349 
Copper,  tin,  zinc  beams,  561,  562 
Copper,   tin,   zinc,   lead,   and   alloys 

in  compression,  391-395 
Copper,  under  repeated  stress,  361 
Core    method    for    general    flexure, 

735-739 

Core  surface  or  section,  732 
Cover     plates     for     plate     girders, 

length  of,  708-710 
Crank  shaft  stresses,  247-253 
Criterion  for  greatest  moment,  84 
Curved  beams,  712-719 
Cylinders^    thick    hollow,     203-223, 

847 

Cylinders,  thin  hollow,  197-201 
Cylinders,  torsion  of,  853-892 


INDEX. 


923 


Deflection  due  to  shearing,  125,  153 
Deflection  in  oblique  flexure,  745-749 
Deflection  in  terms  of  greatest  fibre 

stress,  124 

Deflection  of  beams,  121,  126-131 
Deflection     of     rolled-steel     beams, 

677,  678 

Design  of  columns,  505-520 
Design      of     concrete-steel     beams, 

629-640 
Design    of    concrete-steel    columns, 

653-655 

Diagonal  riveted  joints,  469 
Diameter  of  rivets,  445 
Distribution    of   shear  in   beams   of 

various  sections,  60,  62 
Distribution    of    stress    in    riveted 

joints,  437 
Division  of  loading  between  concrete 

and  steel,  655 

Driving  and  drawing  spikes,  781-786 
Ductility,  286 
Ductility  of  wrought  iron,  302 

E 

Eccentric    loading    of    any    surface, 

725-735 
Effect  of  chemical  elements  on  steel, 

343 
Effect  of  low  temperatures  on  steel, 

333-335 
Effect  of  shpp  manipulation  on  steel, 

339 

Efficiency  of  riveted  joint,  454-461 
Elastic  limit,  5,  282 
Elastic  limit  of  wrought  iron,  298 
Elasticity,  i,  4 
Elasticity,  modulus  of,  4,  281 
Ellipse  of  inertia,  478,  480 
Ellipse  of  strain,  43 
Ellipse  of  stress,  26,  33 
Ellipsoid  of  strain,  42 


Ellipsoid  of  stress,  36,  40 

Elliptical  cylinder,  torsion  of,  186-188, 

54L  863 

End  shear  in  bent  beams,  68 
Equilibrium  and  motion,   equations 

of  internal,  820-846 
Euler's  formula,  169 
Expansion  and  contraction  (thermal) 

of  mortar,  concrete,  and  stone,  377 
Eye-bars  of  steel,  314-327 
Eye-bars  subjected  to  bending  and 

tension,  255,  258,  263 


Fatigue  of  metals,  795-806 

Flanged  beams,  659-682 

Flanged  beams  with  equal  flanges, 

665-682 
Flanged  beams  with  unequal  flanges, 

661-665 

Flat  plates,  square,  rectangular,  cir- 
cular, elliptical,  765-774 
Flexure,  common  theory  of,  49 
Flexure  by  oblique  forces,  175 
Flexure,  general  treatment  by  core 

method,  735~739 
Flexure  of  beams,  121-168 
Flexure  of  beams  of  two  materials, 

156 

Flexure  of  curved  beams,  712-719 
Flexure  of  long  columns,  169,  175 
Flexure,  theory  of,  general  formulas, 

897-912 

Flow  of  solids,  809-819 
Flues,  collapse  of,  774-778 
Formula     (column)     of     C.     Shaler 

Smith  for  timber  columns,  531,  532 
Formulas  for  long  columns,  493-505 
Fracture  of  steel,  343 
Fracture  of  wrought  iron,  302,  303 
Freezing  cements  and  mortars,  effect 

of,  375-377 
Friction  of  riveted  joint,  465 


924 


INDEX. 


General  formulae  of  theory  of  flexure, 

99,  897-912 

Girders,  design  of  plate,  683-708 
Gordon's  formula,  474,  481-490 
Granites  in  compression,  421-425 
Graphical  determination  of  bending 

moments,  160 
Greatest  intensity  of  shearing  stress, 

29,  36,  163 

Greatest  stresses  in  beams,  162 
Gun-bronze,  346,  349,  392 

H    • 

Hardening   and   tempering   of   steel, 

336,  337 

Helical  spiral  springs,  750-760 
High   extreme   fibre   stress   in   short 

solid  beams,  556 

Hollow  cylinders,  thick,  203-223,  847 
Hollow  cylinders,  thin,  197 
Hollow  spheres,  thick,  224,  892 
Hollow  spheres,  thin,  201,  202 
Hooke's  Law,  2,  3 
Hooks,    stresses    in    and    design    of, 

719-725 
Hoop  tension,  204,  206 


Ice     in     compression     and     flexure, 

915-918 
Inclination    of    neutral    surface    of 

beam,  122,  123 

Influence  of  time  on  strains,  805 
Intensity  of  stress,  3 
Intermediate  and  end  shear  in  bent 

beams,  68 


Jaws  of  columns,  design  of,  510,  511 
Joints,  pin  connections,  470-473 


Joints,  riveted,  435,  470 
Joints,  welded,  470 


Lateral  strains,  9 

Lattice  bars,  506-508 

Latticed  columns,  506-516 

Launhardt's  formula,  801,  802 

Law,  Hooke's,  2,  3 

Least  work,  method  of,  788 

Length    of    cover   plates    for     plate 

girders,  708-710 

Limestones  in  compression,  421-425 
Limit  of  elasticity,  5,  282 
Lag-screws,  resistance  to  pulling  out, 

784 

Long  colums,ni69,  175,  474-506 
Long  column  formulas,  493-505  t 

M 

Magnesium,  354,  355 

Magnesium  alloys,  355 

Manganese  steel,  344 

Marbles  in  compression,  421-425 

Method  of  least  work,  788 

Moduli  of  elasticity,  relation  be- 
tween, ii 

Modulus  of  elasticity,  4,  281,  552 

Modulus  of  elasticity  for  tension  and 
compression  in  terms  of  shearing 
elasticity,  19,  20 

Modulus  of  elasticity  for  torsion, 
186,  187,  540-542 

Modulus  of  elasticity  of  alloy  beams, 
562 

Modulus  of  elasticity  of  aluminum- 
zinc  beams,  354 

Modulus  of  elasticity  of  cast  iron, 
286-290,  294,  389 

Modulus  of  elasticity  of  concrete,  399 

Modulus  of  elasticity  of  steel,  303- 
308,  390 


INDEX. 


925 


Modulus  of  elasticity  of  timber  in 

tension,  380 
Modulus    of    elasticity    of    wrought 

iron,  297 
Modulus  of  rupture  in  bending  solid 

rectangular    and    circular    beams, 

554-562 

Moisture  in  timber,  effect  of,  426,  427 
Moment,  greatest,  produced  by  con- 
centrations, 83-86 
Moment  in  cantilever,  126,  128 
Moment  of  inertia,  general  treatment 

of,  475-48o 
Moment  of  single  load  at  centre  of 

span,  95,  129 
Moment  of  uniform  load,  80,  81,  96, 

129 
Moment  produced  by  concentrated 

loads,  83 
Moment    produced    by    two    equal 

weights,  76 
Moments  and  shears  in  bent  beams, 

64 
Moments    in    ordinary    continuous 

beams,  131-142,  144-152 
Moments  tabulated  for  plate  girders, 

89,90 

Mortise  holes,  shearing  behind,  786 
Motion,  equations  of,  820-846 

N 

Natural  building  stones,  420-425,  549 
Neutral  axis,  51,  52 
Neutral  axis,  position  of,  in  reinforced 
concrete  beams,  605-608,  610,  6n 
Neutral  curve  for  continuous  beams, 

132-155 

Neutral     curve    for    special     cases, 

126-132 
Neutral  surface,  shearing  in,  61,  63, 

163 

Nickel  steel,  319,  325-328 
Notation  concrete  steel  beams,  600- 

602 


Oblique  or  general  flexure,  739-749 
Orthogonal  stresses,  43 
Oscillations,  torsional,  886 


Pendulum,  torsion,  890 

Permanent  set,  286 

Phoenix-column  section,  488 

Phoenix  columns,  tests  of,  490-496 

Phosphor-bronze,  361 

Phosphor-bronze  wire,  361 

Pine,  white,  in  compression,  432-434 

Pine,  yellow,  in  compression,  427-434 

Pin  connections,  470-473 

Pitch  of  rivets,  446-453 

Pitch   of  rivets  in  flanges   of  plate 

girder,  698-702,  710,  711 
Plane  spiral  springs,  761-765 
Planes     of     resistance     in     oblique 

flexure,  739~745 
Plate  girder,  design  of,  683-708 
Plates,  carrying  capacity  of,  766-774 
Points    of    contraflexure,    136,    138, 

148,  152 

Poisson's  ratio,  10 
Portland  cement  and  cement  mortar 

in  tension,  362-377 
Portland   cement   concrete   in   com- 
pression, 395-409 
Portland-cement  concrete  in  tension, 

362-377 

Principal  moments  of  inertia,  477-480 
Principal  stresses,  23,  24,  26,  27,  40 
Punching,  drilling,  etc.,  of  steel,  339 

R 

Rail-steel,  323 

Reactions  for  bridge  floor  beams,  74 
Reactions  under  continuous  beams, 
112,  114,  118 


926 


INDEX. 


Rectangular  cylinders,  torsion  of, 
869-883 

Reduction  of  resistance  between 
ultimate  and  breaking  point,  285 

Reinforced  concrete  columns,  641-655 

Resilience,  231 

Resilience  of  cast-iron  in  tension,  290 

Resilience  of  flexure,  233 

Resilience  of  steel  in  tension,  311,  312 

Resilience  of  tension  and  compres- 
sion, 232 

Resilience  of  torsion,  240 

Resilience  of  wrought  iron,  299,  300 

Resilience  of  shearing,  236 

Resilience,  total,  due  to  direct 
stresses  and  shearing,  239 

Resisting  capacity  of  ice,  915-918 

Riveted  joints,  435-470 

Riveted  joints,  butt-joints  with 
double  cover  plates,  for  steel, 
436-452 

Riveted  joints,  distribution  of  stress 

in,  437 

Riveted  joints  for  trusses,  468-470 

Riveted  joints  in  angles,  469 

Riveted  joints,  lap-joints,  and  butt- 
joints  with  single  butt-strap,  for 
steel,  436-448 

Riveted  joints,  tests  of  full-sized, 
454-464 

Riveted  steel  in  shearing,  443,  451, 
460 

Rivets,  bearing  capacity  of,  441,  460 

Rivets,  bending  of,  440 

Rivets,  diameter  and  pitch  of,  445 

Rivets,  shear  of,  443,  460 

Rivets,  steel,  324,  451,  460 

Rollers,  resistance  of,  778-781 


Sandstones  in  compression,  420-425 
Section  modulus,  55 
Set,  permanent,  286 


Shear,  first  derivative  of  moment,  65 
Shear,   greatest   caused  by  uniform 

load,  79 

Shearing,  behind  mortise  holes,  786 
Shearing,   greatest  intensity   of,   29, 

36,  163 
Shearing,    modulus   of   elasticity,    5, 

186,  191 

Shearing  stress  in  beams,  57,  165-167 
Shearing  stress  and  strain,   13,   185, 

1 86,  540 
Shearing  in  neutral  surface  of  timber 

beams,  57,  165-167,  57i~574 
Shearing,  ultimate  resistance,  543-551 
Shears  in  bent  beams,  64,  68 
Shears,  single  load  located  at  centre 

of  span,  95 
Shears,  tabulated  for  plate  girders, 

89,  90 

Shears,  uniform  load  on  span,  97 
Short  blocks,  386 
"Short"  test  specimens,  309,  310 
Shrinkage   stresses    in    thick    hollow 

cylinders,  213 
Silica   sand,    Portland    cement,    and 

mortar  in  tension,  370,  371 
Spheres,  thick  hollow,  224-230,  892 
Spheres,  thin  hollow,  201-202 
Spikes,  driving  and  drawing,  781-786 
Spiral  springs,  helical,  750-765 
Spiral  springs,  plane,  761-765 
Spruce  columns,  429-434 
Spruce  in  compression,  439-443 
Steel,  303 

Steel,  annealing,  338 
Steel  castings,  322 
Steel,    change    of    elastic    properties 

under  repeated  stresses,  342 
Steel,   effect  of  high   and  low  tem- 
peratures, 333-335 
Steel,    effect    of    punching,    drilling, 

reaming,  and  shop  processes,  339 
Steel,   effects  of  chemical  elements, 

343 


INDEX. 


927 


Steel,  elastic  limit,  310,  311,  330,  390 

Steel  eye-bars,  314,  316 

Steel,  fracture  of,  343 

Steel,  hardening  and  tempering,  336, 

337 

Steel,  in  compression,  389-391 
Steel,  in  shearing,  545 
Steel,  in  torsion,  190-192,  545 
Steel,  modulus  of  elasticity,  303-308, 

390 

Steel,  nickel,  325-328 
Steel  rails,  323 
Steel  reinforcement  acquires  stress, 

592-598 
Steel     reinforcement,     economic     or 

balanced,  608-613,  617,  618 
Steel,  resilience  of,  311,  312 
Steel  rivets,  324 

Steel,  rolled  flanged  beams,  669-682 
Steel  shapes  and  plates,  315,  317,  319 
Steel  short  solid  beams,  558 
Steel,  ultimate  tensile  resistance,  305, 

333 

Steel  wire,  320,  321 
Stone  beams,  586,  587 
Stones,     natural,     in     compression, 

420-425 

Stones,  natural,  in  shearing,  549 
Straight-line  formula   for    columns, 

494-504 
Strain,  i,  2,  4 

Strains,  influence  of  time  on,  805 
Stress,  I,  2,  3,  4 
Stress,  intensity  of,  3 
Stress  parallel  to  one  plane,  21 
Stress-strain  curve,  6 
Stress-strain  curves  for  cast  iron,  288 
Stresses  at  any  point  in  beam,  162 
Stresses,    expressions   for   tangential 

and  direct,  820-826 
Stresses  of  tension  and  compression, 

resolution  of,  7,  8 
Structural  steel,  classes  of,  303 
Suddenly  applied  loads,  242,  243 


Temperature,  effect  of  high,  334,  335 

Temperature,  effect  of  low,  333 

Tempering  of  steel,  336,  337 

Tensile  stress,  281 

Terra  cotta  and  columns,  415,  416, 
419 

Tests  of  riveted  joints,  454-464 

Tests  of  steel  angle  and  other  col- 
umns, 496-503 

Tests  of  wrought-iron  Phoenix  col- 
umns, 490-496 

Theorem  of  three  moments,  102,  109, 
in,  114 

Theory  of  flexure,  general  formulae, 
99,  897 

Thermal  expansion  and  contraction 
of  mortars,  concrete,  and  stone, 

377-379 

Thick  hollow  cylinders,  203-223,  847 
Thick  hollow  spheres,  224,  892 
Thin  hollow  cylinders,  197 
Thin  hollow  spheres,  201,  202 
Timber  beams,  563-575 
Timber  columns,  528-539 
Timber  in  compression,  426-434 
Timber    in    shearing    and    torsion, 

547-548 

Timber  in  tension,  379-383 
Tin,  347,  349,  546 
Tin,  alloys  of,  346-356 
Tobin  bronze,  35i~357,  394 
Tobin  bronze  in  compression,  394, 395 
Tobin's  alloy,  346-357 
Torsion,  182,  196,  540,  853 
Torsion,     combined    with-   bending, 

246-253 
Torsion,        general        observations, 

186-188,  885 
Torsion,   greatest  shear    in    circular 

sections,  186-188,  541,  885 
Torsion,  greatest  shear  in  elliptical 

sections,  186-188,  541,  865 


928 


INDEX. 


Torsion,  greatest  shear  in  rectangular 
sections,  186-188,  541,  880-883 

Torsion,  greatest  shear  in  triangular 
sections,  868 

Torsion  in  equilibrium,  182-196,  540, 

853 
Torsion  of  circular  sections,  182-196, 

541,  884 
Torsion  of  elliptical  sections,  186-188, 

541,  863 
Torsion  of  rectangular  sections,  186- 

188,  541,  869 

Torsion  of  triangular  sections,  866 
Torsion  oscillations,  886 
Torsion  pendulum,  890 
Torsion  (twisting)  moment  in  terms 

of  H.P.,  188,  189 
Tresca's  experiments,  flow  of  solids, 

810 
Tresca's  hypotheses,  flow  of  solids, 

811 

U 

Ultimate  resistance,  285,  543 

Ultimate  resistance  affected  by  high 
and  low  temperature,  333-335 

Ultimate  resistance  affected  by  re- 
peated stressing,  361,  795-806 

Ultimate  resistance  of  cast-iron  in 
tension,  292,  294 

Ultimate  resistance  of  steel  in  ten- 
sion, 303-346 

Ultimate  resistance  of  wrought  iron, 
295-303,  387 

V 
Vanadium  steel,  328-333 

W 

Web  reinforcement  in  concrete-steel 

beams,  620-629 
Weight  of  concrete,  372 


Welded  joints,  470 

Weyrauch's  formula,  80 1,  803 

White-oak  columns,  529 

White  oak  in  compression,  429,  432, 

434 

White-pine  columns,  528-539 

White  pine  in  compression,  432-434 

Wire,  steel,  320,  321 

Wohler's  experiments,  796-799 

Wohler's  law,  795-796 

Work  expended  in  producing  strains, 

231 
Working    stresses    in    concrete    steel 

beams,  629 
Working    stresses    in    concrete    steel 

columns,  650 

Wrought  iron,  295-303,  387 
Wrought-iron      bars,      diagram      of 

strains,  298 

Wrought-iron  beams,  680-682 
Wrought  iron,  ductility  and  resilience 

of,  297-300 

Wrought  iron,  fracture  of,  302 
Wrought  iron  in    compression,  387, 

388 

Wrought  iron,  in  shearing,  543 
Wrought  iron,  in  torsion,  192,  543 
Wrought  iron,  modulus  of  elasticity, 

296,  387 

Wrought-iron,  short  solid  beams,  557 

Wrought   iron,    ultimate    resistance, 

and  elastic  limit,  and  yield  point, 

297,  388 


Yield-point,  7,  284 
Yield-point  of  wrought  iron,  297 
Yellow-pine  columns,  528-539 
Yellow  pine  in  compression,  427-434 


Zinc,  346-362,  546 

Zinc,  alloys.of,  193-1 95,  346-362,  546 


14  DAY  USE 

RETURN  TO  DESK  FROM  WHICH  BORROWED 

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